src/HOL/Power.thy
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bootstrap Num.thy before Power.thy; move lemmas about powers into Power.thy
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(*  Title:      HOL/Power.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1997  University of Cambridge
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*)
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header {* Exponentiation *}
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theory Power
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imports Num
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begin
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subsection {* Powers for Arbitrary Monoids *}
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class power = one + times
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begin
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primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80) where
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    power_0: "a ^ 0 = 1"
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  | power_Suc: "a ^ Suc n = a * a ^ n"
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notation (latex output)
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  power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
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notation (HTML output)
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  power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
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end
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context monoid_mult
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begin
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subclass power .
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lemma power_one [simp]:
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  "1 ^ n = 1"
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  by (induct n) simp_all
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lemma power_one_right [simp]:
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  "a ^ 1 = a"
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  by simp
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lemma power_commutes:
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  "a ^ n * a = a * a ^ n"
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  by (induct n) (simp_all add: mult_assoc)
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lemma power_Suc2:
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  "a ^ Suc n = a ^ n * a"
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  by (simp add: power_commutes)
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lemma power_add:
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  "a ^ (m + n) = a ^ m * a ^ n"
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  by (induct m) (simp_all add: algebra_simps)
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lemma power_mult:
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  "a ^ (m * n) = (a ^ m) ^ n"
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  by (induct n) (simp_all add: power_add)
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end
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context comm_monoid_mult
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begin
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lemma power_mult_distrib:
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  "(a * b) ^ n = (a ^ n) * (b ^ n)"
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  by (induct n) (simp_all add: mult_ac)
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end
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context semiring_numeral
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begin
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lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
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  by (simp only: sqr_conv_mult numeral_mult)
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lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
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  by (induct l, simp_all only: numeral_class.numeral.simps pow.simps
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    numeral_sqr numeral_mult power_add power_one_right)
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lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
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  by (rule numeral_pow [symmetric])
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end
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context semiring_1
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begin
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lemma of_nat_power:
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  "of_nat (m ^ n) = of_nat m ^ n"
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  by (induct n) (simp_all add: of_nat_mult)
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lemma power_zero_numeral [simp]: "(0::'a) ^ numeral k = 0"
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  by (cases "numeral k :: nat", 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 {* 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: nat_number(4) power_Suc power_minus_Bit0 mult_minus_left)
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lemma power_neg_numeral_Bit0 [simp]:
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  "neg_numeral k ^ numeral (Num.Bit0 l) = numeral (Num.pow k (Num.Bit0 l))"
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  by (simp only: neg_numeral_def power_minus_Bit0 power_numeral)
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lemma power_neg_numeral_Bit1 [simp]:
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  "neg_numeral k ^ numeral (Num.Bit1 l) = neg_numeral (Num.pow k (Num.Bit1 l))"
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  by (simp only: neg_numeral_def power_minus_Bit1 power_numeral pow.simps)
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end
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context linordered_semidom
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begin
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lemma zero_less_power [simp]:
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  "0 < a \<Longrightarrow> 0 < a ^ n"
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  by (induct n) (simp_all add: mult_pos_pos)
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lemma zero_le_power [simp]:
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  "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
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  by (induct n) (simp_all add: mult_nonneg_nonneg)
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lemma one_le_power[simp]:
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  "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
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  apply (induct n)
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  apply simp_all
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  apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
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  apply (simp_all add: order_trans [OF zero_le_one])
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  done
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lemma power_gt1_lemma:
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  assumes gt1: "1 < a"
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  shows "1 < a * a ^ n"
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proof -
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  from gt1 have "0 \<le> a"
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    by (fact order_trans [OF zero_le_one less_imp_le])
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  have "1 * 1 < a * 1" using gt1 by simp
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  also have "\<dots> \<le> a * a ^ n" using gt1
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    by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le
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        zero_le_one order_refl)
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  finally show ?thesis by simp
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qed
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lemma power_gt1:
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  "1 < a \<Longrightarrow> 1 < a ^ Suc n"
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  by (simp add: power_gt1_lemma)
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lemma one_less_power [simp]:
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  "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
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  by (cases n) (simp_all add: power_gt1_lemma)
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lemma power_le_imp_le_exp:
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  assumes gt1: "1 < a"
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  shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
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proof (induct m arbitrary: n)
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  case 0
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  show ?case by simp
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next
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  case (Suc m)
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  show ?case
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  proof (cases n)
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    case 0
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    with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
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    with gt1 show ?thesis
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      by (force simp only: power_gt1_lemma
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          not_less [symmetric])
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  next
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    case (Suc n)
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    with Suc.prems Suc.hyps show ?thesis
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      by (force dest: mult_left_le_imp_le
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          simp add: less_trans [OF zero_less_one gt1])
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  qed
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qed
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text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
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lemma power_inject_exp [simp]:
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  "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
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  by (force simp add: order_antisym power_le_imp_le_exp)
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text{*Can relax the first premise to @{term "0<a"} in the case of the
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natural numbers.*}
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lemma power_less_imp_less_exp:
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  "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
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  by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
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    power_le_imp_le_exp)
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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)
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    (auto intro: mult_mono order_trans [of 0 a b])
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lemma power_strict_mono [rule_format]:
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  "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
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  by (induct n)
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   (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
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text{*Lemma for @{text power_strict_decreasing}*}
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lemma power_Suc_less:
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  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
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  by (induct n)
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    (auto simp add: mult_strict_left_mono)
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lemma power_strict_decreasing [rule_format]:
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  "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
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proof (induct N)
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  case 0 then show ?case by simp
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next
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  case (Suc N) then show ?case 
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  apply (auto simp add: power_Suc_less less_Suc_eq)
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  apply (subgoal_tac "a * a^N < 1 * a^n")
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  apply simp
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  apply (rule mult_strict_mono) apply auto
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  done
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qed
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text{*Proof resembles that of @{text power_strict_decreasing}*}
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lemma power_decreasing [rule_format]:
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  "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
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proof (induct N)
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  case 0 then show ?case by simp
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next
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  case (Suc N) then show ?case 
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  apply (auto simp add: le_Suc_eq)
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  apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
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  apply (rule mult_mono) apply auto
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  done
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qed
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lemma power_Suc_less_one:
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  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
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  using power_strict_decreasing [of 0 "Suc n" a] by simp
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text{*Proof again resembles that of @{text power_strict_decreasing}*}
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lemma power_increasing [rule_format]:
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  "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
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proof (induct N)
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  case 0 then show ?case by simp
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next
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  case (Suc N) then show ?case 
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  apply (auto simp add: le_Suc_eq)
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  apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
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  apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
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  done
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qed
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text{*Lemma for @{text power_strict_increasing}*}
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lemma power_less_power_Suc:
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  "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
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  by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
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lemma power_strict_increasing [rule_format]:
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  "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
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proof (induct N)
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  case 0 then show ?case by simp
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next
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  case (Suc N) then show ?case 
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  apply (auto simp add: power_less_power_Suc less_Suc_eq)
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  apply (subgoal_tac "1 * a^n < a * a^N", simp)
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  apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
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  done
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qed
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lemma power_increasing_iff [simp]:
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  "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
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  by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
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lemma power_strict_increasing_iff [simp]:
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  "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
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by (blast intro: power_less_imp_less_exp power_strict_increasing) 
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lemma power_le_imp_le_base:
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  assumes le: "a ^ Suc n \<le> b ^ Suc n"
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parents: 30960
diff changeset
   329
    and ynonneg: "0 \<le> b"
648d02b124d8 cleaned up Power theory
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parents: 30960
diff changeset
   330
  shows "a \<le> b"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   331
proof (rule ccontr)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   332
  assume "~ a \<le> b"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   333
  then have "b < a" by (simp only: linorder_not_le)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   334
  then have "b ^ Suc n < a ^ Suc n"
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 39438
diff changeset
   335
    by (simp only: assms power_strict_mono)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   336
  from le and this show False
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   337
    by (simp add: linorder_not_less [symmetric])
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   338
qed
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   339
22853
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   340
lemma power_less_imp_less_base:
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   341
  assumes less: "a ^ n < b ^ n"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   342
  assumes nonneg: "0 \<le> b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   343
  shows "a < b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   344
proof (rule contrapos_pp [OF less])
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   345
  assume "~ a < b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   346
  hence "b \<le> a" by (simp only: linorder_not_less)
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   347
  hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
30996
648d02b124d8 cleaned up Power theory
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parents: 30960
diff changeset
   348
  thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
22853
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   349
qed
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   350
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   351
lemma power_inject_base:
30996
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haftmann
parents: 30960
diff changeset
   352
  "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
648d02b124d8 cleaned up Power theory
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parents: 30960
diff changeset
   353
by (blast intro: power_le_imp_le_base antisym eq_refl sym)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   354
22955
48dc37776d1e add lemma power_eq_imp_eq_base
huffman
parents: 22853
diff changeset
   355
lemma power_eq_imp_eq_base:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   356
  "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   357
  by (cases n) (simp_all del: power_Suc, rule power_inject_base)
22955
48dc37776d1e add lemma power_eq_imp_eq_base
huffman
parents: 22853
diff changeset
   358
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   359
end
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   360
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 33364
diff changeset
   361
context linordered_idom
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   362
begin
29978
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   363
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   364
lemma power_abs:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   365
  "abs (a ^ n) = abs a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   366
  by (induct n) (auto simp add: abs_mult)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   367
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   368
lemma abs_power_minus [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   369
  "abs ((-a) ^ n) = abs (a ^ n)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35028
diff changeset
   370
  by (simp add: power_abs)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   371
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35216
diff changeset
   372
lemma zero_less_power_abs_iff [simp, no_atp]:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   373
  "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   374
proof (induct n)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   375
  case 0 show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   376
next
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   377
  case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
29978
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   378
qed
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   379
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   380
lemma zero_le_power_abs [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   381
  "0 \<le> abs a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   382
  by (rule zero_le_power [OF abs_ge_zero])
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   383
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   384
end
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   385
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   386
context ring_1_no_zero_divisors
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   387
begin
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   388
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   389
lemma field_power_not_zero:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   390
  "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   391
  by (induct n) auto
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   392
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   393
end
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   394
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   395
context division_ring
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   396
begin
29978
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   397
30997
081e825c2218 fixed document generation
haftmann
parents: 30996
diff changeset
   398
text {* FIXME reorient or rename to @{text nonzero_inverse_power} *}
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   399
lemma nonzero_power_inverse:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   400
  "a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   401
  by (induct n)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   402
    (simp_all add: nonzero_inverse_mult_distrib power_commutes field_power_not_zero)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   403
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   404
end
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   405
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   406
context field
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   407
begin
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   408
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   409
lemma nonzero_power_divide:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   410
  "b \<noteq> 0 \<Longrightarrow> (a / b) ^ n = a ^ n / b ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   411
  by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   412
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   413
end
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   414
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   415
lemma power_0_Suc [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   416
  "(0::'a::{power, semiring_0}) ^ Suc n = 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   417
  by simp
30313
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   418
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   419
text{*It looks plausible as a simprule, but its effect can be strange.*}
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   420
lemma power_0_left:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   421
  "0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   422
  by (induct n) simp_all
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   423
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   424
lemma power_eq_0_iff [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   425
  "a ^ n = 0 \<longleftrightarrow>
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   426
     a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,power}) \<and> n \<noteq> 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   427
  by (induct n)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   428
    (auto simp add: no_zero_divisors elim: contrapos_pp)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   429
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   430
lemma (in field) power_diff:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   431
  assumes nz: "a \<noteq> 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   432
  shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   433
  by (induct m n rule: diff_induct) (simp_all add: nz field_power_not_zero)
30313
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   434
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   435
text{*Perhaps these should be simprules.*}
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   436
lemma power_inverse:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   437
  fixes a :: "'a::division_ring_inverse_zero"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   438
  shows "inverse (a ^ n) = inverse a ^ n"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   439
apply (cases "a = 0")
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   440
apply (simp add: power_0_left)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   441
apply (simp add: nonzero_power_inverse)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   442
done (* TODO: reorient or rename to inverse_power *)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   443
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   444
lemma power_one_over:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   445
  "1 / (a::'a::{field_inverse_zero, power}) ^ n =  (1 / a) ^ n"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   446
  by (simp add: divide_inverse) (rule power_inverse)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   447
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   448
lemma power_divide:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   449
  "(a / b) ^ n = (a::'a::field_inverse_zero) ^ n / b ^ n"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   450
apply (cases "b = 0")
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   451
apply (simp add: power_0_left)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   452
apply (rule nonzero_power_divide)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   453
apply assumption
30313
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   454
done
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   455
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   456
30960
fec1a04b7220 power operation defined generic
haftmann
parents: 30730
diff changeset
   457
subsection {* Exponentiation for the Natural Numbers *}
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   458
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   459
lemma nat_one_le_power [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   460
  "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   461
  by (rule one_le_power [of i n, unfolded One_nat_def])
23305
8ae6f7b0903b add lemma of_nat_power
huffman
parents: 23183
diff changeset
   462
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   463
lemma nat_zero_less_power_iff [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   464
  "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   465
  by (induct n) auto
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   466
30056
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   467
lemma nat_power_eq_Suc_0_iff [simp]: 
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   468
  "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   469
  by (induct m) auto
30056
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   470
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   471
lemma power_Suc_0 [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   472
  "Suc 0 ^ n = Suc 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   473
  by simp
30056
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   474
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   475
text{*Valid for the naturals, but what if @{text"0<i<1"}?
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   476
Premises cannot be weakened: consider the case where @{term "i=0"},
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   477
@{term "m=1"} and @{term "n=0"}.*}
21413
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   478
lemma nat_power_less_imp_less:
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   479
  assumes nonneg: "0 < (i\<Colon>nat)"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   480
  assumes less: "i ^ m < i ^ n"
21413
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   481
  shows "m < n"
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   482
proof (cases "i = 1")
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   483
  case True with less power_one [where 'a = nat] show ?thesis by simp
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   484
next
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   485
  case False with nonneg have "1 < i" by auto
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   486
  from power_strict_increasing_iff [OF this] less show ?thesis ..
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   487
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   488
33274
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   489
lemma power_dvd_imp_le:
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   490
  "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   491
  apply (rule power_le_imp_le_exp, assumption)
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   492
  apply (erule dvd_imp_le, simp)
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   493
  done
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   494
31155
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   495
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   496
subsection {* Code generator tweak *}
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   497
45231
d85a2fdc586c replacing code_inline by code_unfold, removing obsolete code_unfold, code_inline del now that the ancient code generator is removed
bulwahn
parents: 41550
diff changeset
   498
lemma power_power_power [code]:
31155
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   499
  "power = power.power (1::'a::{power}) (op *)"
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   500
  unfolding power_def power.power_def ..
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   501
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   502
declare power.power.simps [code]
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   503
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   504
code_modulename SML
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   505
  Power Arith
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   506
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   507
code_modulename OCaml
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   508
  Power Arith
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   509
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   510
code_modulename Haskell
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   511
  Power Arith
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   512
3390
0c7625196d95 New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff changeset
   513
end