src/HOL/Power.thy
author blanchet
Wed Mar 04 10:45:52 2009 +0100 (2009-03-04)
changeset 30240 5b25fee0362c
parent 29608 564ea783ace8
child 30242 aea5d7fa7ef5
permissions -rw-r--r--
Merge.
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(*  Title:      HOL/Power.thy
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    ID:         $Id$
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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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class power =
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  fixes power :: "'a \<Rightarrow> nat \<Rightarrow> 'a"            (infixr "^" 80)
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subsection{*Powers for Arbitrary Monoids*}
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class recpower = monoid_mult + power +
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  assumes power_0 [simp]: "a ^ 0       = 1"
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  assumes power_Suc:      "a ^ Suc n = a * (a ^ n)"
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lemma power_0_Suc [simp]: "(0::'a::{recpower,semiring_0}) ^ (Suc n) = 0"
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  by (simp add: power_Suc)
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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 add: power_Suc)
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lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a"
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  unfolding One_nat_def by (simp add: power_Suc)
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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: power_Suc 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_Suc 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: power_Suc 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_Suc 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: power_Suc 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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apply (induct "n")
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apply (simp_all add: power_Suc zero_less_one mult_pos_pos)
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done
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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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apply (simp add: order_le_less)
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apply (erule disjE)
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apply (simp_all add: zero_less_one power_0_left)
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done
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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 add: power_Suc)
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apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
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apply (simp_all add: zero_le_one 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 power_Suc)
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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 power_Suc)
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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 add: power_Suc)
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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: power_Suc 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 add: power_Suc)
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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 power_Suc
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                      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: power_Suc zero_neq_one [THEN not_sym] 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: power_Suc 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: power_Suc abs_mult)
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done
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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 add: zero_less_one)
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next
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  case (Suc n)
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    show ?case by (auto simp add: prems power_Suc zero_less_mult_iff
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      abs_zero)
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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 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: power_Suc 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 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: zero_less_one 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: power_Suc  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 simp add: zero_le_one)
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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: power_Suc 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: power_Suc 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 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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  "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x \<le> b ^ y) = (x \<le> y)"
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by (blast intro: power_le_imp_le_exp power_increasing order_less_imp_le) 
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lemma power_strict_increasing_iff [simp]:
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  "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x < b ^ y) = (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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    and ynonneg: "(0::'a::{ordered_semidom,recpower}) \<le> b"
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shows "a \<le> b"
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proof (rule ccontr)
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  assume "~ a \<le> b"
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  then have "b < a" by (simp only: linorder_not_le)
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  then have "b ^ Suc n < a ^ Suc n"
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    by (simp only: prems power_strict_mono)
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  from le and this show "False"
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    by (simp add: linorder_not_less [symmetric])
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qed
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lemma power_less_imp_less_base:
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  fixes a b :: "'a::{ordered_semidom,recpower}"
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  assumes less: "a ^ n < b ^ n"
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  assumes nonneg: "0 \<le> b"
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  shows "a < b"
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proof (rule contrapos_pp [OF less])
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  assume "~ a < b"
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  hence "b \<le> a" by (simp only: linorder_not_less)
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  hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
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  thus "~ a ^ n < b ^ n" by (simp only: linorder_not_less)
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qed
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lemma power_inject_base:
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     "[| a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b |]
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      ==> a = (b::'a::{ordered_semidom,recpower})"
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by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym)
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lemma power_eq_imp_eq_base:
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  fixes a b :: "'a::{ordered_semidom,recpower}"
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  shows "\<lbrakk>a ^ n = b ^ n; 0 \<le> a; 0 \<le> b; 0 < n\<rbrakk> \<Longrightarrow> a = b"
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by (cases n, simp_all, rule power_inject_base)
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text {* The divides relation *}
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lemma le_imp_power_dvd:
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  fixes a :: "'a::{comm_semiring_1,recpower}"
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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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   341
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   342
lemma power_le_dvd:
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  fixes a b :: "'a::{comm_semiring_1,recpower}"
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  shows "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])
blanchet@30240
   346
paulson@14348
   347
paulson@14348
   348
subsection{*Exponentiation for the Natural Numbers*}
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haftmann@25836
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instantiation nat :: recpower
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begin
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   352
haftmann@25836
   353
primrec power_nat where
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  "p ^ 0 = (1\<Colon>nat)"
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  | "p ^ (Suc n) = (p\<Colon>nat) * (p ^ n)"
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   356
haftmann@25836
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instance proof
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  fix z n :: nat
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  show "z^0 = 1" by simp
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   360
  show "z^(Suc n) = z * (z^n)" by simp
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qed
paulson@14348
   362
haftmann@25836
   363
end
haftmann@25836
   364
huffman@23305
   365
lemma of_nat_power:
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  "of_nat (m ^ n) = (of_nat m::'a::{semiring_1,recpower}) ^ n"
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   367
by (induct n, simp_all add: power_Suc of_nat_mult)
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   368
blanchet@30240
   369
lemma nat_one_le_power [simp]: "Suc 0 \<le> i ==> Suc 0 \<le> i^n"
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by (rule one_le_power [of i n, unfolded One_nat_def])
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   371
nipkow@25162
   372
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
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by (induct "n", auto)
paulson@14348
   374
blanchet@30240
   375
lemma nat_power_eq_Suc_0_iff [simp]: 
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  "((x::nat)^m = Suc 0) = (m = 0 | x = Suc 0)"
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   377
by (induct_tac m, auto)
blanchet@30240
   378
blanchet@30240
   379
lemma power_Suc_0[simp]: "(Suc 0)^n = Suc 0"
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   380
by simp
blanchet@30240
   381
paulson@14348
   382
text{*Valid for the naturals, but what if @{text"0<i<1"}?
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   383
Premises cannot be weakened: consider the case where @{term "i=0"},
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   384
@{term "m=1"} and @{term "n=0"}.*}
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   385
lemma nat_power_less_imp_less:
haftmann@21413
   386
  assumes nonneg: "0 < (i\<Colon>nat)"
haftmann@21413
   387
  assumes less: "i^m < i^n"
haftmann@21413
   388
  shows "m < n"
haftmann@21413
   389
proof (cases "i = 1")
haftmann@21413
   390
  case True with less power_one [where 'a = nat] show ?thesis by simp
haftmann@21413
   391
next
haftmann@21413
   392
  case False with nonneg have "1 < i" by auto
haftmann@21413
   393
  from power_strict_increasing_iff [OF this] less show ?thesis ..
haftmann@21413
   394
qed
paulson@14348
   395
ballarin@17149
   396
lemma power_diff:
ballarin@17149
   397
  assumes nz: "a ~= 0"
ballarin@17149
   398
  shows "n <= m ==> (a::'a::{recpower, field}) ^ (m-n) = (a^m) / (a^n)"
ballarin@17149
   399
  by (induct m n rule: diff_induct)
ballarin@17149
   400
    (simp_all add: power_Suc nonzero_mult_divide_cancel_left nz)
ballarin@17149
   401
ballarin@17149
   402
paulson@14348
   403
text{*ML bindings for the general exponentiation theorems*}
paulson@14348
   404
ML
paulson@14348
   405
{*
paulson@14348
   406
val power_0 = thm"power_0";
paulson@14348
   407
val power_Suc = thm"power_Suc";
paulson@14348
   408
val power_0_Suc = thm"power_0_Suc";
paulson@14348
   409
val power_0_left = thm"power_0_left";
paulson@14348
   410
val power_one = thm"power_one";
paulson@14348
   411
val power_one_right = thm"power_one_right";
paulson@14348
   412
val power_add = thm"power_add";
paulson@14348
   413
val power_mult = thm"power_mult";
paulson@14348
   414
val power_mult_distrib = thm"power_mult_distrib";
paulson@14348
   415
val zero_less_power = thm"zero_less_power";
paulson@14348
   416
val zero_le_power = thm"zero_le_power";
paulson@14348
   417
val one_le_power = thm"one_le_power";
paulson@14348
   418
val gt1_imp_ge0 = thm"gt1_imp_ge0";
paulson@14348
   419
val power_gt1_lemma = thm"power_gt1_lemma";
paulson@14348
   420
val power_gt1 = thm"power_gt1";
paulson@14348
   421
val power_le_imp_le_exp = thm"power_le_imp_le_exp";
paulson@14348
   422
val power_inject_exp = thm"power_inject_exp";
paulson@14348
   423
val power_less_imp_less_exp = thm"power_less_imp_less_exp";
paulson@14348
   424
val power_mono = thm"power_mono";
paulson@14348
   425
val power_strict_mono = thm"power_strict_mono";
paulson@14348
   426
val power_eq_0_iff = thm"power_eq_0_iff";
nipkow@25134
   427
val field_power_eq_0_iff = thm"power_eq_0_iff";
paulson@14348
   428
val field_power_not_zero = thm"field_power_not_zero";
paulson@14348
   429
val power_inverse = thm"power_inverse";
paulson@14353
   430
val nonzero_power_divide = thm"nonzero_power_divide";
paulson@14353
   431
val power_divide = thm"power_divide";
paulson@14348
   432
val power_abs = thm"power_abs";
paulson@14353
   433
val zero_less_power_abs_iff = thm"zero_less_power_abs_iff";
paulson@14353
   434
val zero_le_power_abs = thm "zero_le_power_abs";
paulson@14348
   435
val power_minus = thm"power_minus";
paulson@14348
   436
val power_Suc_less = thm"power_Suc_less";
paulson@14348
   437
val power_strict_decreasing = thm"power_strict_decreasing";
paulson@14348
   438
val power_decreasing = thm"power_decreasing";
paulson@14348
   439
val power_Suc_less_one = thm"power_Suc_less_one";
paulson@14348
   440
val power_increasing = thm"power_increasing";
paulson@14348
   441
val power_strict_increasing = thm"power_strict_increasing";
paulson@14348
   442
val power_le_imp_le_base = thm"power_le_imp_le_base";
paulson@14348
   443
val power_inject_base = thm"power_inject_base";
paulson@14348
   444
*}
wenzelm@14577
   445
paulson@14348
   446
text{*ML bindings for the remaining theorems*}
paulson@14348
   447
ML
paulson@14348
   448
{*
paulson@14348
   449
val nat_one_le_power = thm"nat_one_le_power";
paulson@14348
   450
val nat_power_less_imp_less = thm"nat_power_less_imp_less";
paulson@14348
   451
val nat_zero_less_power_iff = thm"nat_zero_less_power_iff";
paulson@14348
   452
*}
paulson@3390
   453
paulson@3390
   454
end