src/HOL/Power.thy
author wenzelm
Mon Apr 25 16:09:26 2016 +0200 (2016-04-25)
changeset 63040 eb4ddd18d635
parent 62481 b5d8e57826df
child 63417 c184ec919c70
permissions -rw-r--r--
eliminated old 'def';
tuned comments;
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(*  Title:      HOL/Power.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1997  University of Cambridge
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*)
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section \<open>Exponentiation\<close>
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theory Power
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imports Num
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begin
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subsection \<open>Powers for Arbitrary Monoids\<close>
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class power = one + times
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begin
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primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a"  (infixr "^" 80)
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where
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  power_0: "a ^ 0 = 1"
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| power_Suc: "a ^ Suc n = a * a ^ n"
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notation (latex output)
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  power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
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text \<open>Special syntax for squares.\<close>
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abbreviation power2 :: "'a \<Rightarrow> 'a"  ("(_\<^sup>2)" [1000] 999)
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  where "x\<^sup>2 \<equiv> x ^ 2"
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end
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context monoid_mult
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begin
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subclass power .
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lemma power_one [simp]:
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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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  define g where "g x = f x - 1" for x
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  with Suc have "n = g x" by simp
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  with Suc have "times x ^^ g x = times (x ^ g x)" by simp
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  moreover from Suc g_def have "f x = g x + 1" by simp
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  ultimately show ?case by (simp add: power_add funpow_add fun_eq_iff mult.assoc)
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qed
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lemma power_commuting_commutes:
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  assumes "x * y = y * x"
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  shows "x ^ n * y = y * x ^n"
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proof (induct n)
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  case (Suc n)
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  have "x ^ Suc n * y = x ^ n * y * x"
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    by (subst power_Suc2) (simp add: assms ac_simps)
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  also have "\<dots> = y * x ^ Suc n"
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    unfolding Suc power_Suc2
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    by (simp add: ac_simps)
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  finally show ?case .
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qed simp
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lemma power_minus_mult:
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  "0 < n \<Longrightarrow> a ^ (n - 1) * a = a ^ n"
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  by (simp add: power_commutes split add: nat_diff_split)
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end
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context comm_monoid_mult
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begin
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lemma power_mult_distrib [field_simps]:
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  "(a * b) ^ n = (a ^ n) * (b ^ n)"
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  by (induct n) (simp_all add: ac_simps)
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end
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text\<open>Extract constant factors from powers\<close>
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declare power_mult_distrib [where a = "numeral w" for w, simp]
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declare power_mult_distrib [where b = "numeral w" for w, simp]
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lemma power_add_numeral [simp]:
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  fixes a :: "'a :: monoid_mult"
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  shows "a^numeral m * a^numeral n = a^numeral (m + n)"
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  by (simp add: power_add [symmetric])
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lemma power_add_numeral2 [simp]:
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  fixes a :: "'a :: monoid_mult"
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  shows "a^numeral m * (a^numeral n * b) = a^numeral (m + n) * b"
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  by (simp add: mult.assoc [symmetric])
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lemma power_mult_numeral [simp]:
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  fixes a :: "'a :: monoid_mult"
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  shows"(a^numeral m)^numeral n = a^numeral (m * n)"
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  by (simp only: numeral_mult power_mult)
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context semiring_numeral
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begin
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lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
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  by (simp only: sqr_conv_mult numeral_mult)
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lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
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  by (induct l, simp_all only: numeral_class.numeral.simps pow.simps
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    numeral_sqr numeral_mult power_add power_one_right)
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lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
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  by (rule numeral_pow [symmetric])
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end
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context semiring_1
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begin
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lemma of_nat_power [simp]:
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  "of_nat (m ^ n) = of_nat m ^ n"
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  by (induct n) (simp_all add: of_nat_mult)
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lemma zero_power:
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  "0 < n \<Longrightarrow> 0 ^ n = 0"
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  by (cases n) simp_all
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lemma power_zero_numeral [simp]:
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  "0 ^ numeral k = 0"
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  by (simp add: numeral_eq_Suc)
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lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
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  by (rule power_zero_numeral)
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lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
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  by (rule power_one)
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lemma power_0_Suc [simp]:
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  "0 ^ Suc n = 0"
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  by simp
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text\<open>It looks plausible as a simprule, but its effect can be strange.\<close>
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lemma power_0_left:
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  "0 ^ n = (if n = 0 then 1 else 0)"
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  by (cases n) simp_all
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end
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context comm_semiring_1
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begin
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text \<open>The divides relation\<close>
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lemma le_imp_power_dvd:
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  assumes "m \<le> n" shows "a ^ m dvd a ^ n"
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proof
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  have "a ^ n = a ^ (m + (n - m))"
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    using \<open>m \<le> n\<close> by simp
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  also have "\<dots> = a ^ m * a ^ (n - m)"
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    by (rule power_add)
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  finally show "a ^ n = a ^ m * a ^ (n - m)" .
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qed
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lemma power_le_dvd:
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  "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
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  by (rule dvd_trans [OF le_imp_power_dvd])
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lemma dvd_power_same:
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  "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
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  by (induct n) (auto simp add: mult_dvd_mono)
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lemma dvd_power_le:
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  "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
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  by (rule power_le_dvd [OF dvd_power_same])
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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 semiring_1_no_zero_divisors
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begin
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subclass power .
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lemma power_eq_0_iff [simp]:
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  "a ^ n = 0 \<longleftrightarrow> a = 0 \<and> n > 0"
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  by (induct n) auto
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lemma power_not_zero:
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  "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
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  by (induct n) auto
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lemma zero_eq_power2 [simp]:
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  "a\<^sup>2 = 0 \<longleftrightarrow> a = 0"
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  unfolding power2_eq_square by simp
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end
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context ring_1
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begin
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lemma power_minus:
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  "(- a) ^ n = (- 1) ^ n * a ^ n"
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proof (induct n)
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  case 0 show ?case by simp
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next
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  case (Suc n) then show ?case
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    by (simp del: power_Suc add: power_Suc2 mult.assoc)
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qed
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lemma power_minus': "NO_MATCH 1 x \<Longrightarrow> (-x) ^ n = (-1)^n * x ^ n"
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  by (rule power_minus)
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lemma power_minus_Bit0:
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  "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
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  by (induct k, simp_all only: numeral_class.numeral.simps power_add
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    power_one_right mult_minus_left mult_minus_right minus_minus)
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lemma power_minus_Bit1:
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  "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
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  by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)
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lemma power2_minus [simp]:
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  "(- a)\<^sup>2 = a\<^sup>2"
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  by (fact power_minus_Bit0)
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lemma power_minus1_even [simp]:
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  "(- 1) ^ (2*n) = 1"
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proof (induct n)
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  case 0 show ?case by simp
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next
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  case (Suc n) then show ?case by (simp add: power_add power2_eq_square)
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qed
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lemma power_minus1_odd:
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  "(- 1) ^ Suc (2*n) = -1"
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  by simp
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lemma power_minus_even [simp]:
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  "(-a) ^ (2*n) = a ^ (2*n)"
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  by (simp add: power_minus [of a])
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end
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context ring_1_no_zero_divisors
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begin
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lemma power2_eq_1_iff:
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  "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
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  using square_eq_1_iff [of a] by (simp add: power2_eq_square)
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end
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context idom
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begin
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lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y"
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  unfolding power2_eq_square by (rule square_eq_iff)
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end
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context algebraic_semidom
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begin
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lemma div_power:
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  assumes "b dvd a"
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  shows "(a div b) ^ n = a ^ n div b ^ n"
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  using assms by (induct n) (simp_all add: div_mult_div_if_dvd dvd_power_same)
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lemma is_unit_power_iff:
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  "is_unit (a ^ n) \<longleftrightarrow> is_unit a \<or> n = 0"
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  by (induct n) (auto simp add: is_unit_mult_iff)
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end
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context normalization_semidom
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begin
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lemma normalize_power:
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  "normalize (a ^ n) = normalize a ^ n"
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  by (induct n) (simp_all add: normalize_mult)
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lemma unit_factor_power:
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  "unit_factor (a ^ n) = unit_factor a ^ n"
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  by (induct n) (simp_all add: unit_factor_mult)
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end
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context division_ring
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begin
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text\<open>Perhaps these should be simprules.\<close>
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lemma power_inverse [field_simps, divide_simps]:
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  "inverse a ^ n = inverse (a ^ n)"
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proof (cases "a = 0")
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  case True then show ?thesis by (simp add: power_0_left)
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next
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  case False then have "inverse (a ^ n) = inverse a ^ n"
haftmann@60867
   354
    by (induct n) (simp_all add: nonzero_inverse_mult_distrib power_commutes)
haftmann@60867
   355
  then show ?thesis by simp
haftmann@60867
   356
qed
huffman@47192
   357
haftmann@60867
   358
lemma power_one_over [field_simps, divide_simps]:
haftmann@60867
   359
  "(1 / a) ^ n = 1 / a ^ n"
haftmann@60867
   360
  using power_inverse [of a] by (simp add: divide_inverse)
haftmann@60867
   361
lp15@61649
   362
end
huffman@47192
   363
huffman@47192
   364
context field
huffman@47192
   365
begin
huffman@47192
   366
haftmann@60867
   367
lemma power_diff:
haftmann@60867
   368
  assumes nz: "a \<noteq> 0"
haftmann@60867
   369
  shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
haftmann@60867
   370
  by (induct m n rule: diff_induct) (simp_all add: nz power_not_zero)
huffman@47192
   371
haftmann@60867
   372
lemma power_divide [field_simps, divide_simps]:
haftmann@60867
   373
  "(a / b) ^ n = a ^ n / b ^ n"
haftmann@60867
   374
  by (induct n) simp_all
haftmann@60867
   375
huffman@47192
   376
end
huffman@47192
   377
huffman@47192
   378
wenzelm@60758
   379
subsection \<open>Exponentiation on ordered types\<close>
huffman@47192
   380
haftmann@35028
   381
context linordered_semidom
haftmann@30996
   382
begin
haftmann@30996
   383
haftmann@30996
   384
lemma zero_less_power [simp]:
haftmann@30996
   385
  "0 < a \<Longrightarrow> 0 < a ^ n"
nipkow@56544
   386
  by (induct n) simp_all
haftmann@30996
   387
haftmann@30996
   388
lemma zero_le_power [simp]:
haftmann@30996
   389
  "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
nipkow@56536
   390
  by (induct n) simp_all
paulson@14348
   391
huffman@47241
   392
lemma power_mono:
huffman@47241
   393
  "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
huffman@47241
   394
  by (induct n) (auto intro: mult_mono order_trans [of 0 a b])
huffman@47241
   395
huffman@47241
   396
lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
huffman@47241
   397
  using power_mono [of 1 a n] by simp
huffman@47241
   398
huffman@47241
   399
lemma power_le_one: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1"
huffman@47241
   400
  using power_mono [of a 1 n] by simp
paulson@14348
   401
paulson@14348
   402
lemma power_gt1_lemma:
haftmann@30996
   403
  assumes gt1: "1 < a"
haftmann@30996
   404
  shows "1 < a * a ^ n"
paulson@14348
   405
proof -
haftmann@30996
   406
  from gt1 have "0 \<le> a"
haftmann@30996
   407
    by (fact order_trans [OF zero_le_one less_imp_le])
haftmann@30996
   408
  have "1 * 1 < a * 1" using gt1 by simp
haftmann@30996
   409
  also have "\<dots> \<le> a * a ^ n" using gt1
wenzelm@60758
   410
    by (simp only: mult_mono \<open>0 \<le> a\<close> one_le_power order_less_imp_le
wenzelm@14577
   411
        zero_le_one order_refl)
wenzelm@14577
   412
  finally show ?thesis by simp
paulson@14348
   413
qed
paulson@14348
   414
haftmann@30996
   415
lemma power_gt1:
haftmann@30996
   416
  "1 < a \<Longrightarrow> 1 < a ^ Suc n"
haftmann@30996
   417
  by (simp add: power_gt1_lemma)
huffman@24376
   418
haftmann@30996
   419
lemma one_less_power [simp]:
haftmann@30996
   420
  "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
haftmann@30996
   421
  by (cases n) (simp_all add: power_gt1_lemma)
paulson@14348
   422
paulson@14348
   423
lemma power_le_imp_le_exp:
haftmann@30996
   424
  assumes gt1: "1 < a"
haftmann@30996
   425
  shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
haftmann@30996
   426
proof (induct m arbitrary: n)
paulson@14348
   427
  case 0
wenzelm@14577
   428
  show ?case by simp
paulson@14348
   429
next
paulson@14348
   430
  case (Suc m)
wenzelm@14577
   431
  show ?case
wenzelm@14577
   432
  proof (cases n)
wenzelm@14577
   433
    case 0
haftmann@30996
   434
    with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
wenzelm@14577
   435
    with gt1 show ?thesis
wenzelm@14577
   436
      by (force simp only: power_gt1_lemma
haftmann@30996
   437
          not_less [symmetric])
wenzelm@14577
   438
  next
wenzelm@14577
   439
    case (Suc n)
haftmann@30996
   440
    with Suc.prems Suc.hyps show ?thesis
wenzelm@14577
   441
      by (force dest: mult_left_le_imp_le
haftmann@30996
   442
          simp add: less_trans [OF zero_less_one gt1])
wenzelm@14577
   443
  qed
paulson@14348
   444
qed
paulson@14348
   445
lp15@61649
   446
lemma of_nat_zero_less_power_iff [simp]:
lp15@61649
   447
  "of_nat x ^ n > 0 \<longleftrightarrow> x > 0 \<or> n = 0"
lp15@61649
   448
  by (induct n) auto
lp15@61649
   449
wenzelm@61799
   450
text\<open>Surely we can strengthen this? It holds for \<open>0<a<1\<close> too.\<close>
paulson@14348
   451
lemma power_inject_exp [simp]:
haftmann@30996
   452
  "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
wenzelm@14577
   453
  by (force simp add: order_antisym power_le_imp_le_exp)
paulson@14348
   454
wenzelm@60758
   455
text\<open>Can relax the first premise to @{term "0<a"} in the case of the
wenzelm@60758
   456
natural numbers.\<close>
paulson@14348
   457
lemma power_less_imp_less_exp:
haftmann@30996
   458
  "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
haftmann@30996
   459
  by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
haftmann@30996
   460
    power_le_imp_le_exp)
paulson@14348
   461
paulson@14348
   462
lemma power_strict_mono [rule_format]:
haftmann@30996
   463
  "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
haftmann@30996
   464
  by (induct n)
haftmann@30996
   465
   (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
paulson@14348
   466
wenzelm@61799
   467
text\<open>Lemma for \<open>power_strict_decreasing\<close>\<close>
paulson@14348
   468
lemma power_Suc_less:
haftmann@30996
   469
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
haftmann@30996
   470
  by (induct n)
haftmann@30996
   471
    (auto simp add: mult_strict_left_mono)
paulson@14348
   472
haftmann@30996
   473
lemma power_strict_decreasing [rule_format]:
haftmann@30996
   474
  "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
haftmann@30996
   475
proof (induct N)
haftmann@30996
   476
  case 0 then show ?case by simp
haftmann@30996
   477
next
lp15@61649
   478
  case (Suc N) then show ?case
haftmann@30996
   479
  apply (auto simp add: power_Suc_less less_Suc_eq)
haftmann@30996
   480
  apply (subgoal_tac "a * a^N < 1 * a^n")
haftmann@30996
   481
  apply simp
haftmann@30996
   482
  apply (rule mult_strict_mono) apply auto
haftmann@30996
   483
  done
haftmann@30996
   484
qed
paulson@14348
   485
wenzelm@61799
   486
text\<open>Proof resembles that of \<open>power_strict_decreasing\<close>\<close>
haftmann@30996
   487
lemma power_decreasing [rule_format]:
haftmann@30996
   488
  "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
haftmann@30996
   489
proof (induct N)
haftmann@30996
   490
  case 0 then show ?case by simp
haftmann@30996
   491
next
lp15@61649
   492
  case (Suc N) then show ?case
haftmann@30996
   493
  apply (auto simp add: le_Suc_eq)
haftmann@30996
   494
  apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
haftmann@30996
   495
  apply (rule mult_mono) apply auto
haftmann@30996
   496
  done
haftmann@30996
   497
qed
paulson@14348
   498
paulson@14348
   499
lemma power_Suc_less_one:
haftmann@30996
   500
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
haftmann@30996
   501
  using power_strict_decreasing [of 0 "Suc n" a] by simp
paulson@14348
   502
wenzelm@61799
   503
text\<open>Proof again resembles that of \<open>power_strict_decreasing\<close>\<close>
haftmann@30996
   504
lemma power_increasing [rule_format]:
haftmann@30996
   505
  "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
haftmann@30996
   506
proof (induct N)
haftmann@30996
   507
  case 0 then show ?case by simp
haftmann@30996
   508
next
lp15@61649
   509
  case (Suc N) then show ?case
haftmann@30996
   510
  apply (auto simp add: le_Suc_eq)
haftmann@30996
   511
  apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
haftmann@30996
   512
  apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
haftmann@30996
   513
  done
haftmann@30996
   514
qed
paulson@14348
   515
wenzelm@61799
   516
text\<open>Lemma for \<open>power_strict_increasing\<close>\<close>
paulson@14348
   517
lemma power_less_power_Suc:
haftmann@30996
   518
  "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
haftmann@30996
   519
  by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
paulson@14348
   520
haftmann@30996
   521
lemma power_strict_increasing [rule_format]:
haftmann@30996
   522
  "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
haftmann@30996
   523
proof (induct N)
haftmann@30996
   524
  case 0 then show ?case by simp
haftmann@30996
   525
next
lp15@61649
   526
  case (Suc N) then show ?case
haftmann@30996
   527
  apply (auto simp add: power_less_power_Suc less_Suc_eq)
haftmann@30996
   528
  apply (subgoal_tac "1 * a^n < a * a^N", simp)
haftmann@30996
   529
  apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
haftmann@30996
   530
  done
haftmann@30996
   531
qed
paulson@14348
   532
nipkow@25134
   533
lemma power_increasing_iff [simp]:
haftmann@30996
   534
  "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
haftmann@30996
   535
  by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
paulson@15066
   536
paulson@15066
   537
lemma power_strict_increasing_iff [simp]:
haftmann@30996
   538
  "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
lp15@61649
   539
by (blast intro: power_less_imp_less_exp power_strict_increasing)
paulson@15066
   540
paulson@14348
   541
lemma power_le_imp_le_base:
haftmann@30996
   542
  assumes le: "a ^ Suc n \<le> b ^ Suc n"
haftmann@30996
   543
    and ynonneg: "0 \<le> b"
haftmann@30996
   544
  shows "a \<le> b"
nipkow@25134
   545
proof (rule ccontr)
nipkow@25134
   546
  assume "~ a \<le> b"
nipkow@25134
   547
  then have "b < a" by (simp only: linorder_not_le)
nipkow@25134
   548
  then have "b ^ Suc n < a ^ Suc n"
wenzelm@41550
   549
    by (simp only: assms power_strict_mono)
haftmann@30996
   550
  from le and this show False
nipkow@25134
   551
    by (simp add: linorder_not_less [symmetric])
nipkow@25134
   552
qed
wenzelm@14577
   553
huffman@22853
   554
lemma power_less_imp_less_base:
huffman@22853
   555
  assumes less: "a ^ n < b ^ n"
huffman@22853
   556
  assumes nonneg: "0 \<le> b"
huffman@22853
   557
  shows "a < b"
huffman@22853
   558
proof (rule contrapos_pp [OF less])
huffman@22853
   559
  assume "~ a < b"
huffman@22853
   560
  hence "b \<le> a" by (simp only: linorder_not_less)
huffman@22853
   561
  hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
haftmann@30996
   562
  thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
huffman@22853
   563
qed
huffman@22853
   564
paulson@14348
   565
lemma power_inject_base:
haftmann@30996
   566
  "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
haftmann@30996
   567
by (blast intro: power_le_imp_le_base antisym eq_refl sym)
paulson@14348
   568
huffman@22955
   569
lemma power_eq_imp_eq_base:
haftmann@30996
   570
  "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
haftmann@30996
   571
  by (cases n) (simp_all del: power_Suc, rule power_inject_base)
huffman@22955
   572
haftmann@62347
   573
lemma power_eq_iff_eq_base:
haftmann@62347
   574
  "0 < n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a ^ n = b ^ n \<longleftrightarrow> a = b"
haftmann@62347
   575
  using power_eq_imp_eq_base [of a n b] by auto
haftmann@62347
   576
huffman@47192
   577
lemma power2_le_imp_le:
wenzelm@53015
   578
  "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
huffman@47192
   579
  unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
huffman@47192
   580
huffman@47192
   581
lemma power2_less_imp_less:
wenzelm@53015
   582
  "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
huffman@47192
   583
  by (rule power_less_imp_less_base)
huffman@47192
   584
huffman@47192
   585
lemma power2_eq_imp_eq:
wenzelm@53015
   586
  "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
huffman@47192
   587
  unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
huffman@47192
   588
haftmann@62347
   589
lemma power_Suc_le_self:
haftmann@62347
   590
  shows "0 \<le> a \<Longrightarrow> a \<le> 1 \<Longrightarrow> a ^ Suc n \<le> a"
haftmann@62347
   591
  using power_decreasing [of 1 "Suc n" a] by simp
haftmann@62347
   592
huffman@47192
   593
end
huffman@47192
   594
huffman@47192
   595
context linordered_ring_strict
huffman@47192
   596
begin
huffman@47192
   597
huffman@47192
   598
lemma sum_squares_eq_zero_iff:
huffman@47192
   599
  "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   600
  by (simp add: add_nonneg_eq_0_iff)
huffman@47192
   601
huffman@47192
   602
lemma sum_squares_le_zero_iff:
huffman@47192
   603
  "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   604
  by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
huffman@47192
   605
huffman@47192
   606
lemma sum_squares_gt_zero_iff:
huffman@47192
   607
  "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
huffman@47192
   608
  by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
huffman@47192
   609
haftmann@30996
   610
end
haftmann@30996
   611
haftmann@35028
   612
context linordered_idom
haftmann@30996
   613
begin
huffman@29978
   614
wenzelm@61944
   615
lemma power_abs: "\<bar>a ^ n\<bar> = \<bar>a\<bar> ^ n"
haftmann@30996
   616
  by (induct n) (auto simp add: abs_mult)
haftmann@30996
   617
wenzelm@61944
   618
lemma abs_power_minus [simp]: "\<bar>(-a) ^ n\<bar> = \<bar>a ^ n\<bar>"
huffman@35216
   619
  by (simp add: power_abs)
haftmann@30996
   620
wenzelm@61944
   621
lemma zero_less_power_abs_iff [simp]: "0 < \<bar>a\<bar> ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
haftmann@30996
   622
proof (induct n)
haftmann@30996
   623
  case 0 show ?case by simp
haftmann@30996
   624
next
haftmann@30996
   625
  case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
huffman@29978
   626
qed
huffman@29978
   627
wenzelm@61944
   628
lemma zero_le_power_abs [simp]: "0 \<le> \<bar>a\<bar> ^ n"
haftmann@30996
   629
  by (rule zero_le_power [OF abs_ge_zero])
haftmann@30996
   630
huffman@47192
   631
lemma zero_le_power2 [simp]:
wenzelm@53015
   632
  "0 \<le> a\<^sup>2"
huffman@47192
   633
  by (simp add: power2_eq_square)
huffman@47192
   634
huffman@47192
   635
lemma zero_less_power2 [simp]:
wenzelm@53015
   636
  "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
huffman@47192
   637
  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
huffman@47192
   638
huffman@47192
   639
lemma power2_less_0 [simp]:
wenzelm@53015
   640
  "\<not> a\<^sup>2 < 0"
huffman@47192
   641
  by (force simp add: power2_eq_square mult_less_0_iff)
huffman@47192
   642
haftmann@58787
   643
lemma power2_less_eq_zero_iff [simp]:
haftmann@58787
   644
  "a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
haftmann@58787
   645
  by (simp add: le_less)
haftmann@58787
   646
wenzelm@61944
   647
lemma abs_power2 [simp]: "\<bar>a\<^sup>2\<bar> = a\<^sup>2"
huffman@47192
   648
  by (simp add: power2_eq_square abs_mult abs_mult_self)
huffman@47192
   649
wenzelm@61944
   650
lemma power2_abs [simp]: "\<bar>a\<bar>\<^sup>2 = a\<^sup>2"
huffman@47192
   651
  by (simp add: power2_eq_square abs_mult_self)
huffman@47192
   652
huffman@47192
   653
lemma odd_power_less_zero:
huffman@47192
   654
  "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
huffman@47192
   655
proof (induct n)
huffman@47192
   656
  case 0
huffman@47192
   657
  then show ?case by simp
huffman@47192
   658
next
huffman@47192
   659
  case (Suc n)
huffman@47192
   660
  have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
haftmann@57514
   661
    by (simp add: ac_simps power_add power2_eq_square)
huffman@47192
   662
  thus ?case
huffman@47192
   663
    by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
huffman@47192
   664
qed
haftmann@30996
   665
huffman@47192
   666
lemma odd_0_le_power_imp_0_le:
huffman@47192
   667
  "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
huffman@47192
   668
  using odd_power_less_zero [of a n]
lp15@61649
   669
    by (force simp add: linorder_not_less [symmetric])
huffman@47192
   670
huffman@47192
   671
lemma zero_le_even_power'[simp]:
huffman@47192
   672
  "0 \<le> a ^ (2*n)"
huffman@47192
   673
proof (induct n)
huffman@47192
   674
  case 0
huffman@47192
   675
    show ?case by simp
huffman@47192
   676
next
huffman@47192
   677
  case (Suc n)
lp15@61649
   678
    have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
haftmann@57514
   679
      by (simp add: ac_simps power_add power2_eq_square)
huffman@47192
   680
    thus ?case
huffman@47192
   681
      by (simp add: Suc zero_le_mult_iff)
huffman@47192
   682
qed
haftmann@30996
   683
huffman@47192
   684
lemma sum_power2_ge_zero:
wenzelm@53015
   685
  "0 \<le> x\<^sup>2 + y\<^sup>2"
huffman@47192
   686
  by (intro add_nonneg_nonneg zero_le_power2)
huffman@47192
   687
huffman@47192
   688
lemma not_sum_power2_lt_zero:
wenzelm@53015
   689
  "\<not> x\<^sup>2 + y\<^sup>2 < 0"
huffman@47192
   690
  unfolding not_less by (rule sum_power2_ge_zero)
huffman@47192
   691
huffman@47192
   692
lemma sum_power2_eq_zero_iff:
wenzelm@53015
   693
  "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   694
  unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
huffman@47192
   695
huffman@47192
   696
lemma sum_power2_le_zero_iff:
wenzelm@53015
   697
  "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   698
  by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
huffman@47192
   699
huffman@47192
   700
lemma sum_power2_gt_zero_iff:
wenzelm@53015
   701
  "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
huffman@47192
   702
  unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
haftmann@30996
   703
lp15@59865
   704
lemma abs_le_square_iff:
lp15@59865
   705
   "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> x\<^sup>2 \<le> y\<^sup>2"
lp15@59865
   706
proof
lp15@59865
   707
  assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
lp15@59865
   708
  then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono, simp)
lp15@59865
   709
  then show "x\<^sup>2 \<le> y\<^sup>2" by simp
lp15@59865
   710
next
lp15@59865
   711
  assume "x\<^sup>2 \<le> y\<^sup>2"
lp15@59865
   712
  then show "\<bar>x\<bar> \<le> \<bar>y\<bar>"
lp15@59865
   713
    by (auto intro!: power2_le_imp_le [OF _ abs_ge_zero])
lp15@59865
   714
qed
lp15@59865
   715
wenzelm@61944
   716
lemma abs_square_le_1:"x\<^sup>2 \<le> 1 \<longleftrightarrow> \<bar>x\<bar> \<le> 1"
lp15@59865
   717
  using abs_le_square_iff [of x 1]
lp15@59865
   718
  by simp
lp15@59865
   719
wenzelm@61944
   720
lemma abs_square_eq_1: "x\<^sup>2 = 1 \<longleftrightarrow> \<bar>x\<bar> = 1"
lp15@59865
   721
  by (auto simp add: abs_if power2_eq_1_iff)
lp15@61649
   722
wenzelm@61944
   723
lemma abs_square_less_1: "x\<^sup>2 < 1 \<longleftrightarrow> \<bar>x\<bar> < 1"
lp15@59865
   724
  using  abs_square_eq_1 [of x] abs_square_le_1 [of x]
lp15@59865
   725
  by (auto simp add: le_less)
lp15@59865
   726
haftmann@30996
   727
end
haftmann@30996
   728
huffman@29978
   729
wenzelm@60758
   730
subsection \<open>Miscellaneous rules\<close>
paulson@14348
   731
haftmann@60867
   732
lemma (in linordered_semidom) self_le_power:
haftmann@60867
   733
  "1 \<le> a \<Longrightarrow> 0 < n \<Longrightarrow> a \<le> a ^ n"
haftmann@60867
   734
  using power_increasing [of 1 n a] power_one_right [of a] by auto
lp15@55718
   735
haftmann@60867
   736
lemma (in power) power_eq_if:
haftmann@60867
   737
  "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
huffman@47255
   738
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
   739
haftmann@58787
   740
lemma (in comm_semiring_1) power2_sum:
haftmann@58787
   741
  "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
huffman@47192
   742
  by (simp add: algebra_simps power2_eq_square mult_2_right)
haftmann@30996
   743
haftmann@58787
   744
lemma (in comm_ring_1) power2_diff:
haftmann@58787
   745
  "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
haftmann@58787
   746
  by (simp add: algebra_simps power2_eq_square mult_2_right)
haftmann@30996
   747
lp15@60974
   748
lemma (in comm_ring_1) power2_commute:
lp15@60974
   749
  "(x - y)\<^sup>2 = (y - x)\<^sup>2"
lp15@60974
   750
  by (simp add: algebra_simps power2_eq_square)
lp15@60974
   751
wenzelm@60758
   752
text \<open>Simprules for comparisons where common factors can be cancelled.\<close>
huffman@47255
   753
huffman@47255
   754
lemmas zero_compare_simps =
huffman@47255
   755
    add_strict_increasing add_strict_increasing2 add_increasing
lp15@61649
   756
    zero_le_mult_iff zero_le_divide_iff
lp15@61649
   757
    zero_less_mult_iff zero_less_divide_iff
lp15@61649
   758
    mult_le_0_iff divide_le_0_iff
lp15@61649
   759
    mult_less_0_iff divide_less_0_iff
huffman@47255
   760
    zero_le_power2 power2_less_0
huffman@47255
   761
nipkow@30313
   762
wenzelm@60758
   763
subsection \<open>Exponentiation for the Natural Numbers\<close>
wenzelm@14577
   764
haftmann@30996
   765
lemma nat_one_le_power [simp]:
haftmann@30996
   766
  "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
haftmann@30996
   767
  by (rule one_le_power [of i n, unfolded One_nat_def])
huffman@23305
   768
haftmann@30996
   769
lemma nat_zero_less_power_iff [simp]:
haftmann@30996
   770
  "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
haftmann@30996
   771
  by (induct n) auto
paulson@14348
   772
lp15@61649
   773
lemma nat_power_eq_Suc_0_iff [simp]:
haftmann@30996
   774
  "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
haftmann@30996
   775
  by (induct m) auto
nipkow@30056
   776
haftmann@30996
   777
lemma power_Suc_0 [simp]:
haftmann@30996
   778
  "Suc 0 ^ n = Suc 0"
haftmann@30996
   779
  by simp
nipkow@30056
   780
wenzelm@61799
   781
text\<open>Valid for the naturals, but what if \<open>0<i<1\<close>?
paulson@14348
   782
Premises cannot be weakened: consider the case where @{term "i=0"},
wenzelm@60758
   783
@{term "m=1"} and @{term "n=0"}.\<close>
haftmann@21413
   784
lemma nat_power_less_imp_less:
wenzelm@61076
   785
  assumes nonneg: "0 < (i::nat)"
haftmann@30996
   786
  assumes less: "i ^ m < i ^ n"
haftmann@21413
   787
  shows "m < n"
haftmann@21413
   788
proof (cases "i = 1")
haftmann@21413
   789
  case True with less power_one [where 'a = nat] show ?thesis by simp
haftmann@21413
   790
next
haftmann@21413
   791
  case False with nonneg have "1 < i" by auto
haftmann@21413
   792
  from power_strict_increasing_iff [OF this] less show ?thesis ..
haftmann@21413
   793
qed
paulson@14348
   794
haftmann@33274
   795
lemma power_dvd_imp_le:
haftmann@33274
   796
  "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
haftmann@33274
   797
  apply (rule power_le_imp_le_exp, assumption)
haftmann@33274
   798
  apply (erule dvd_imp_le, simp)
haftmann@33274
   799
  done
haftmann@33274
   800
haftmann@51263
   801
lemma power2_nat_le_eq_le:
haftmann@51263
   802
  fixes m n :: nat
wenzelm@53015
   803
  shows "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
haftmann@51263
   804
  by (auto intro: power2_le_imp_le power_mono)
haftmann@51263
   805
haftmann@51263
   806
lemma power2_nat_le_imp_le:
haftmann@51263
   807
  fixes m n :: nat
wenzelm@53015
   808
  assumes "m\<^sup>2 \<le> n"
haftmann@51263
   809
  shows "m \<le> n"
haftmann@54249
   810
proof (cases m)
haftmann@54249
   811
  case 0 then show ?thesis by simp
haftmann@54249
   812
next
haftmann@54249
   813
  case (Suc k)
haftmann@54249
   814
  show ?thesis
haftmann@54249
   815
  proof (rule ccontr)
haftmann@54249
   816
    assume "\<not> m \<le> n"
haftmann@54249
   817
    then have "n < m" by simp
haftmann@54249
   818
    with assms Suc show False
haftmann@60867
   819
      by (simp add: power2_eq_square)
haftmann@54249
   820
  qed
haftmann@54249
   821
qed
haftmann@51263
   822
wenzelm@60758
   823
subsubsection \<open>Cardinality of the Powerset\<close>
traytel@55096
   824
traytel@55096
   825
lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
traytel@55096
   826
  unfolding UNIV_bool by simp
traytel@55096
   827
traytel@55096
   828
lemma card_Pow: "finite A \<Longrightarrow> card (Pow A) = 2 ^ card A"
traytel@55096
   829
proof (induct rule: finite_induct)
lp15@61649
   830
  case empty
traytel@55096
   831
    show ?case by auto
traytel@55096
   832
next
traytel@55096
   833
  case (insert x A)
lp15@61649
   834
  then have "inj_on (insert x) (Pow A)"
traytel@55096
   835
    unfolding inj_on_def by (blast elim!: equalityE)
lp15@61649
   836
  then have "card (Pow A) + card (insert x ` Pow A) = 2 * 2 ^ card A"
traytel@55096
   837
    by (simp add: mult_2 card_image Pow_insert insert.hyps)
traytel@55096
   838
  then show ?case using insert
traytel@55096
   839
    apply (simp add: Pow_insert)
traytel@55096
   840
    apply (subst card_Un_disjoint, auto)
traytel@55096
   841
    done
traytel@55096
   842
qed
traytel@55096
   843
haftmann@57418
   844
wenzelm@60758
   845
subsection \<open>Code generator tweak\<close>
haftmann@31155
   846
haftmann@52435
   847
code_identifier
haftmann@52435
   848
  code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
   849
paulson@3390
   850
end