src/HOL/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 Nat
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begin
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subsection {* Powers for Arbitrary Monoids *}
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class recpower = monoid_mult
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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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end
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lemma power_0_Suc [simp]: "(0::'a::{recpower,semiring_0}) ^ (Suc n) = 0"
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  by simp
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text{*It looks plausible as a simprule, but its effect can be strange.*}
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lemma power_0_left: "0^n = (if n=0 then 1 else (0::'a::{recpower,semiring_0}))"
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  by (induct n) simp_all
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lemma power_one [simp]: "1^n = (1::'a::recpower)"
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  by (induct n) simp_all
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lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a"
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  unfolding One_nat_def by simp
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lemma power_commutes: "(a::'a::recpower) ^ n * a = a * a ^ n"
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  by (induct n) (simp_all add: mult_assoc)
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lemma power_Suc2: "(a::'a::recpower) ^ Suc n = a ^ n * a"
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  by (simp add: power_commutes)
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lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)"
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  by (induct m) (simp_all add: mult_ac)
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lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n"
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  by (induct n) (simp_all add: power_add)
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lemma power_mult_distrib: "((a::'a::{recpower,comm_monoid_mult}) * b) ^ n = (a^n) * (b^n)"
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  by (induct n) (simp_all add: mult_ac)
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lemma zero_less_power[simp]:
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     "0 < (a::'a::{ordered_semidom,recpower}) ==> 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::'a::{ordered_semidom,recpower}) ==> 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::'a::{ordered_semidom,recpower}) ==> 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 gt1_imp_ge0: "1 < a ==> 0 \<le> (a::'a::ordered_semidom)"
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  by (simp add: order_trans [OF zero_le_one order_less_imp_le])
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lemma power_gt1_lemma:
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  assumes gt1: "1 < (a::'a::{ordered_semidom,recpower})"
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  shows "1 < a * a^n"
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proof -
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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 gt1_imp_ge0 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 one_less_power[simp]:
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  "\<lbrakk>1 < (a::'a::{ordered_semidom,recpower}); 0 < n\<rbrakk> \<Longrightarrow> 1 < a ^ n"
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by (cases n, simp_all add: power_gt1_lemma)
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lemma power_gt1:
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     "1 < (a::'a::{ordered_semidom,recpower}) ==> 1 < a ^ (Suc n)"
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by (simp add: power_gt1_lemma)
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lemma power_le_imp_le_exp:
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  assumes gt1: "(1::'a::{recpower,ordered_semidom}) < a"
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  shows "!!n. a^m \<le> a^n ==> m \<le> n"
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proof (induct m)
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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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    from prems 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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          linorder_not_less [symmetric])
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  next
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    case (Suc n)
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    from prems show ?thesis
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      by (force dest: mult_left_le_imp_le
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          simp add: order_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::'a::{ordered_semidom,recpower}) ==> (a^m = a^n) = (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::{recpower,ordered_semidom}) < a; a^m < a^n |] ==> m < n"
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by (simp add: order_less_le [of m n] order_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; (0::'a::{recpower,ordered_semidom}) \<le> a|] ==> a^n \<le> b^n"
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apply (induct "n")
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apply simp_all
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apply (auto intro: mult_mono order_trans [of 0 a b])
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done
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lemma power_strict_mono [rule_format]:
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     "[|a < b; (0::'a::{recpower,ordered_semidom}) \<le> a|]
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      ==> 0 < n --> a^n < b^n"
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apply (induct "n")
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apply (auto simp add: mult_strict_mono order_le_less_trans [of 0 a b])
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done
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lemma power_eq_0_iff [simp]:
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  "(a^n = 0) \<longleftrightarrow>
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   (a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,recpower}) & n\<noteq>0)"
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apply (induct "n")
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apply (auto simp add: no_zero_divisors)
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done
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lemma field_power_not_zero:
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  "a \<noteq> (0::'a::{ring_1_no_zero_divisors,recpower}) ==> a^n \<noteq> 0"
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by force
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lemma nonzero_power_inverse:
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  fixes a :: "'a::{division_ring,recpower}"
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  shows "a \<noteq> 0 ==> inverse (a ^ n) = (inverse a) ^ n"
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apply (induct "n")
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apply (auto simp add: nonzero_inverse_mult_distrib power_commutes)
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done (* TODO: reorient or rename to nonzero_inverse_power *)
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text{*Perhaps these should be simprules.*}
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lemma power_inverse:
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  fixes a :: "'a::{division_ring,division_by_zero,recpower}"
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  shows "inverse (a ^ n) = (inverse a) ^ n"
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apply (cases "a = 0")
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apply (simp add: power_0_left)
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apply (simp add: nonzero_power_inverse)
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done (* TODO: reorient or rename to inverse_power *)
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lemma power_one_over: "1 / (a::'a::{field,division_by_zero,recpower})^n = 
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    (1 / a)^n"
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apply (simp add: divide_inverse)
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apply (rule power_inverse)
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done
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lemma nonzero_power_divide:
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    "b \<noteq> 0 ==> (a/b) ^ n = ((a::'a::{field,recpower}) ^ n) / (b ^ n)"
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by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
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lemma power_divide:
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    "(a/b) ^ n = ((a::'a::{field,division_by_zero,recpower}) ^ n / b ^ n)"
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apply (case_tac "b=0", simp add: power_0_left)
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apply (rule nonzero_power_divide)
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apply assumption
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done
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lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,recpower}) ^ n"
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apply (induct "n")
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apply (auto simp add: abs_mult)
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done
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lemma abs_power_minus [simp]:
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  fixes a:: "'a::{ordered_idom,recpower}" shows "abs((-a) ^ n) = abs(a ^ n)"
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  by (simp add: abs_minus_cancel power_abs) 
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lemma zero_less_power_abs_iff [simp,noatp]:
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     "(0 < (abs a)^n) = (a \<noteq> (0::'a::{ordered_idom,recpower}) | n=0)"
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proof (induct "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 n)
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    show ?case by (auto simp add: prems zero_less_mult_iff)
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qed
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lemma zero_le_power_abs [simp]:
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     "(0::'a::{ordered_idom,recpower}) \<le> (abs a)^n"
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by (rule zero_le_power [OF abs_ge_zero])
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lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{ring_1,recpower}) ^ 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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text{*Lemma for @{text power_strict_decreasing}*}
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lemma power_Suc_less:
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     "[|(0::'a::{ordered_semidom,recpower}) < a; a < 1|]
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      ==> a * a^n < a^n"
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apply (induct n)
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apply (auto simp add: mult_strict_left_mono)
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done
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lemma power_strict_decreasing:
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     "[|n < N; 0 < a; a < (1::'a::{ordered_semidom,recpower})|]
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      ==> a^N < a^n"
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apply (erule rev_mp)
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apply (induct "N")
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apply (auto simp add: power_Suc_less less_Suc_eq)
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apply (rename_tac m)
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apply (subgoal_tac "a * a^m < 1 * a^n", simp)
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apply (rule mult_strict_mono)
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apply (auto simp add: order_less_imp_le)
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done
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text{*Proof resembles that of @{text power_strict_decreasing}*}
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lemma power_decreasing:
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     "[|n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semidom,recpower})|]
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      ==> a^N \<le> a^n"
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apply (erule rev_mp)
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apply (induct "N")
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apply (auto simp add: le_Suc_eq)
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apply (rename_tac m)
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apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp)
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apply (rule mult_mono)
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apply auto
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done
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lemma power_Suc_less_one:
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     "[| 0 < a; a < (1::'a::{ordered_semidom,recpower}) |] ==> a ^ Suc n < 1"
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apply (insert power_strict_decreasing [of 0 "Suc n" a], simp)
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done
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text{*Proof again resembles that of @{text power_strict_decreasing}*}
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lemma power_increasing:
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     "[|n \<le> N; (1::'a::{ordered_semidom,recpower}) \<le> a|] ==> a^n \<le> a^N"
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apply (erule rev_mp)
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apply (induct "N")
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apply (auto simp add: le_Suc_eq)
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apply (rename_tac m)
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apply (subgoal_tac "1 * a^n \<le> a * a^m", simp)
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apply (rule mult_mono)
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apply (auto simp add: order_trans [OF zero_le_one])
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done
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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::{ordered_semidom,recpower}) < a ==> a^n < a * a^n"
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apply (induct n)
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apply (auto simp add: mult_strict_left_mono order_less_trans [OF zero_less_one])
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done
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lemma power_strict_increasing:
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     "[|n < N; (1::'a::{ordered_semidom,recpower}) < a|] ==> a^n < a^N"
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apply (erule rev_mp)
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apply (induct "N")
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apply (auto simp add: power_less_power_Suc less_Suc_eq)
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apply (rename_tac m)
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apply (subgoal_tac "1 * a^n < a * a^m", simp)
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apply (rule mult_strict_mono)
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apply (auto simp add: order_less_trans [OF zero_less_one] order_less_imp_le)
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done
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lemma power_increasing_iff [simp]:
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   284
  "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x \<le> b ^ y) = (x \<le> y)"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   285
by (blast intro: power_le_imp_le_exp power_increasing order_less_imp_le) 
15066
d2f2b908e0a4 two new results
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parents: 15004
diff changeset
   286
d2f2b908e0a4 two new results
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parents: 15004
diff changeset
   287
lemma power_strict_increasing_iff [simp]:
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
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parents: 25062
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   288
  "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x < b ^ y) = (x < y)"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   289
by (blast intro: power_less_imp_less_exp power_strict_increasing) 
15066
d2f2b908e0a4 two new results
paulson
parents: 15004
diff changeset
   290
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744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
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   291
lemma power_le_imp_le_base:
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
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   292
assumes le: "a ^ Suc n \<le> b ^ Suc n"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   293
    and ynonneg: "(0::'a::{ordered_semidom,recpower}) \<le> b"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   294
shows "a \<le> b"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   295
proof (rule ccontr)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   296
  assume "~ a \<le> b"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   297
  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
   298
  then have "b ^ Suc n < a ^ Suc n"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   299
    by (simp only: prems power_strict_mono)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   300
  from le and this show "False"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   301
    by (simp add: linorder_not_less [symmetric])
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   302
qed
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dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   303
22853
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   304
lemma power_less_imp_less_base:
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   305
  fixes a b :: "'a::{ordered_semidom,recpower}"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   306
  assumes less: "a ^ n < b ^ n"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   307
  assumes nonneg: "0 \<le> b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   308
  shows "a < b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   309
proof (rule contrapos_pp [OF less])
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   310
  assume "~ a < b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   311
  hence "b \<le> a" by (simp only: linorder_not_less)
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   312
  hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   313
  thus "~ a ^ n < b ^ n" by (simp only: linorder_not_less)
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   314
qed
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   315
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
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   316
lemma power_inject_base:
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   317
     "[| a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b |]
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
   318
      ==> a = (b::'a::{ordered_semidom,recpower})"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   319
by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym)
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   320
22955
48dc37776d1e add lemma power_eq_imp_eq_base
huffman
parents: 22853
diff changeset
   321
lemma power_eq_imp_eq_base:
48dc37776d1e add lemma power_eq_imp_eq_base
huffman
parents: 22853
diff changeset
   322
  fixes a b :: "'a::{ordered_semidom,recpower}"
48dc37776d1e add lemma power_eq_imp_eq_base
huffman
parents: 22853
diff changeset
   323
  shows "\<lbrakk>a ^ n = b ^ n; 0 \<le> a; 0 \<le> b; 0 < n\<rbrakk> \<Longrightarrow> a = b"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30242
diff changeset
   324
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
   325
29978
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   326
text {* The divides relation *}
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   327
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   328
lemma le_imp_power_dvd:
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   329
  fixes a :: "'a::{comm_semiring_1,recpower}"
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   330
  assumes "m \<le> n" shows "a^m dvd a^n"
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   331
proof
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   332
  have "a^n = a^(m + (n - m))"
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   333
    using `m \<le> n` by simp
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   334
  also have "\<dots> = a^m * a^(n - m)"
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   335
    by (rule power_add)
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   336
  finally show "a^n = a^m * a^(n - m)" .
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   337
qed
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   338
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   339
lemma power_le_dvd:
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   340
  fixes a b :: "'a::{comm_semiring_1,recpower}"
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   341
  shows "a^n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a^m dvd b"
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   342
  by (rule dvd_trans [OF le_imp_power_dvd])
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   343
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   344
30313
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   345
lemma dvd_power_same:
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   346
  "(x::'a::{comm_semiring_1,recpower}) dvd y \<Longrightarrow> x^n dvd y^n"
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   347
by (induct n) (auto simp add: mult_dvd_mono)
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   348
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   349
lemma dvd_power_le:
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   350
  "(x::'a::{comm_semiring_1,recpower}) dvd y \<Longrightarrow> m >= n \<Longrightarrow> x^n dvd y^m"
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   351
by(rule power_le_dvd[OF dvd_power_same])
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   352
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   353
lemma dvd_power [simp]:
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   354
  "n > 0 | (x::'a::{comm_semiring_1,recpower}) = 1 \<Longrightarrow> x dvd x^n"
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   355
apply (erule disjE)
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   356
 apply (subgoal_tac "x ^ n = x^(Suc (n - 1))")
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   357
  apply (erule ssubst)
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   358
  apply (subst power_Suc)
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   359
  apply auto
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   360
done
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   361
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   362
30960
fec1a04b7220 power operation defined generic
haftmann
parents: 30730
diff changeset
   363
subsection {* Exponentiation for the Natural Numbers *}
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   364
30960
fec1a04b7220 power operation defined generic
haftmann
parents: 30730
diff changeset
   365
instance nat :: recpower ..
25836
f7771e4f7064 more instantiation
haftmann
parents: 25231
diff changeset
   366
23305
8ae6f7b0903b add lemma of_nat_power
huffman
parents: 23183
diff changeset
   367
lemma of_nat_power:
8ae6f7b0903b add lemma of_nat_power
huffman
parents: 23183
diff changeset
   368
  "of_nat (m ^ n) = (of_nat m::'a::{semiring_1,recpower}) ^ n"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30242
diff changeset
   369
by (induct n, simp_all add: of_nat_mult)
23305
8ae6f7b0903b add lemma of_nat_power
huffman
parents: 23183
diff changeset
   370
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   371
lemma nat_one_le_power [simp]: "Suc 0 \<le> i ==> Suc 0 \<le> i^n"
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   372
by (rule one_le_power [of i n, unfolded One_nat_def])
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   373
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   374
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
21413
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   375
by (induct "n", auto)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   376
30056
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   377
lemma nat_power_eq_Suc_0_iff [simp]: 
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   378
  "((x::nat)^m = Suc 0) = (m = 0 | x = Suc 0)"
30960
fec1a04b7220 power operation defined generic
haftmann
parents: 30730
diff changeset
   379
by (induct m, auto)
30056
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   380
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   381
lemma power_Suc_0[simp]: "(Suc 0)^n = Suc 0"
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   382
by simp
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   383
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   384
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
   385
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
   386
@{term "m=1"} and @{term "n=0"}.*}
21413
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   387
lemma nat_power_less_imp_less:
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   388
  assumes nonneg: "0 < (i\<Colon>nat)"
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   389
  assumes less: "i^m < i^n"
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   390
  shows "m < n"
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   391
proof (cases "i = 1")
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   392
  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
   393
next
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   394
  case False with nonneg have "1 < i" by auto
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   395
  from power_strict_increasing_iff [OF this] less show ?thesis ..
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   396
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   397
17149
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16796
diff changeset
   398
lemma power_diff:
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16796
diff changeset
   399
  assumes nz: "a ~= 0"
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16796
diff changeset
   400
  shows "n <= m ==> (a::'a::{recpower, field}) ^ (m-n) = (a^m) / (a^n)"
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16796
diff changeset
   401
  by (induct m n rule: diff_induct)
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30242
diff changeset
   402
    (simp_all add: nonzero_mult_divide_cancel_left nz)
17149
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16796
diff changeset
   403
3390
0c7625196d95 New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff changeset
   404
end