author  nipkow 
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permissions  rwrr 
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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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linear arithmetic now takes "&" in assumptions apart.
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header{*Exponentiation*} 
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15131  10 
theory Power 
21413  11 
imports Nat 
15131  12 
begin 
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class power = 
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fixes power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80) 
24996  16 

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subsection{*Powers for Arbitrary Monoids*} 
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22390  19 
class recpower = monoid_mult + power + 
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assumes power_Suc [simp]: "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 
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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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15004  30 
lemma power_one [simp]: "1^n = (1::'a::recpower)" 
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by (induct n) simp_all 
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15004  33 
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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15004  42 
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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15004  45 
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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25874  51 
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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25874  55 
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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25874  59 
lemma one_le_power[simp]: 
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"1 \<le> (a::'a::{ordered_semidom,recpower}) ==> 1 \<le> a^n" 
15251  61 
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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14738  67 
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})" 
14577  72 
shows "1 < a * a^n" 
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proof  
14577  74 
have "1*1 < a*1" using gt1 by simp 
75 
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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25874  81 
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" 
14577  91 
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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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 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 (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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14577  173 
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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14577  177 
lemma power_divide: 
15004  178 
"(a/b) ^ n = ((a::'a::{field,division_by_zero,recpower}) ^ n / b ^ n)" 
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179 
apply (case_tac "b=0", simp add: power_0_left) 
14577  180 
apply (rule nonzero_power_divide) 
181 
apply assumption 

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182 
done 
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183 

15004  184 
lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,recpower}) ^ n" 
15251  185 
apply (induct "n") 
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apply (auto simp add: abs_mult) 
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187 
done 
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188 

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lemma zero_less_power_abs_iff [simp,noatp]: 
15004  190 
"(0 < (abs a)^n) = (a \<noteq> (0::'a::{ordered_idom,recpower})  n=0)" 
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191 
proof (induct "n") 
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case 0 
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193 
show ?case by simp 
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194 
next 
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195 
case (Suc n) 
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show ?case by (auto simp add: prems zero_less_mult_iff) 
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197 
qed 
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198 

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199 
lemma zero_le_power_abs [simp]: 
15004  200 
"(0::'a::{ordered_idom,recpower}) \<le> (abs a)^n" 
22957  201 
by (rule zero_le_power [OF abs_ge_zero]) 
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202 

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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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205 
case 0 show ?case by simp 
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206 
next 
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207 
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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210 

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text{*Lemma for @{text power_strict_decreasing}*} 
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lemma power_Suc_less: 
15004  213 
"[(0::'a::{ordered_semidom,recpower}) < a; a < 1] 
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==> a * a^n < a^n" 
15251  215 
apply (induct n) 
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apply (auto simp add: mult_strict_left_mono) 
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217 
done 
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218 

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219 
lemma power_strict_decreasing: 
15004  220 
"[n < N; 0 < a; a < (1::'a::{ordered_semidom,recpower})] 
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==> a^N < a^n" 
14577  222 
apply (erule rev_mp) 
15251  223 
apply (induct "N") 
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224 
apply (auto simp add: power_Suc_less less_Suc_eq) 
14577  225 
apply (rename_tac m) 
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226 
apply (subgoal_tac "a * a^m < 1 * a^n", simp) 
14577  227 
apply (rule mult_strict_mono) 
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228 
apply (auto simp add: order_less_imp_le) 
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229 
done 
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230 

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text{*Proof resembles that of @{text power_strict_decreasing}*} 
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232 
lemma power_decreasing: 
15004  233 
"[n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semidom,recpower})] 
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==> a^N \<le> a^n" 
14577  235 
apply (erule rev_mp) 
15251  236 
apply (induct "N") 
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237 
apply (auto simp add: le_Suc_eq) 
14577  238 
apply (rename_tac m) 
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239 
apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp) 
14577  240 
apply (rule mult_mono) 
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241 
apply auto 
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242 
done 
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243 

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244 
lemma power_Suc_less_one: 
15004  245 
"[ 0 < a; a < (1::'a::{ordered_semidom,recpower}) ] ==> a ^ Suc n < 1" 
14577  246 
apply (insert power_strict_decreasing [of 0 "Suc n" a], simp) 
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247 
done 
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248 

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249 
text{*Proof again resembles that of @{text power_strict_decreasing}*} 
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250 
lemma power_increasing: 
15004  251 
"[n \<le> N; (1::'a::{ordered_semidom,recpower}) \<le> a] ==> a^n \<le> a^N" 
14577  252 
apply (erule rev_mp) 
15251  253 
apply (induct "N") 
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254 
apply (auto simp add: le_Suc_eq) 
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255 
apply (rename_tac m) 
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256 
apply (subgoal_tac "1 * a^n \<le> a * a^m", simp) 
14577  257 
apply (rule mult_mono) 
25874  258 
apply (auto simp add: order_trans [OF zero_le_one]) 
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259 
done 
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260 

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text{*Lemma for @{text power_strict_increasing}*} 
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262 
lemma power_less_power_Suc: 
15004  263 
"(1::'a::{ordered_semidom,recpower}) < a ==> a^n < a * a^n" 
15251  264 
apply (induct n) 
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265 
apply (auto simp add: mult_strict_left_mono order_less_trans [OF zero_less_one]) 
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266 
done 
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267 

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268 
lemma power_strict_increasing: 
15004  269 
"[n < N; (1::'a::{ordered_semidom,recpower}) < a] ==> a^n < a^N" 
14577  270 
apply (erule rev_mp) 
15251  271 
apply (induct "N") 
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272 
apply (auto simp add: power_less_power_Suc less_Suc_eq) 
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273 
apply (rename_tac m) 
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274 
apply (subgoal_tac "1 * a^n < a * a^m", simp) 
14577  275 
apply (rule mult_strict_mono) 
25874  276 
apply (auto simp add: order_less_trans [OF zero_less_one] order_less_imp_le) 
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277 
done 
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278 

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279 
lemma power_increasing_iff [simp]: 
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280 
"1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x \<le> b ^ y) = (x \<le> y)" 
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281 
by (blast intro: power_le_imp_le_exp power_increasing order_less_imp_le) 
15066  282 

283 
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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285 
by (blast intro: power_less_imp_less_exp power_strict_increasing) 
15066  286 

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287 
lemma power_le_imp_le_base: 
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288 
assumes le: "a ^ Suc n \<le> b ^ Suc n" 
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289 
and ynonneg: "(0::'a::{ordered_semidom,recpower}) \<le> b" 
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290 
shows "a \<le> b" 
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291 
proof (rule ccontr) 
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292 
assume "~ a \<le> b" 
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293 
then have "b < a" by (simp only: linorder_not_le) 
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294 
then have "b ^ Suc n < a ^ Suc n" 
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295 
by (simp only: prems power_strict_mono) 
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296 
from le and this show "False" 
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297 
by (simp add: linorder_not_less [symmetric]) 
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298 
qed 
14577  299 

22853  300 
lemma power_less_imp_less_base: 
301 
fixes a b :: "'a::{ordered_semidom,recpower}" 

302 
assumes less: "a ^ n < b ^ n" 

303 
assumes nonneg: "0 \<le> b" 

304 
shows "a < b" 

305 
proof (rule contrapos_pp [OF less]) 

306 
assume "~ a < b" 

307 
hence "b \<le> a" by (simp only: linorder_not_less) 

308 
hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono) 

309 
thus "~ a ^ n < b ^ n" by (simp only: linorder_not_less) 

310 
qed 

311 

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312 
lemma power_inject_base: 
14577  313 
"[ a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b ] 
15004  314 
==> a = (b::'a::{ordered_semidom,recpower})" 
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315 
by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym) 
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316 

22955  317 
lemma power_eq_imp_eq_base: 
318 
fixes a b :: "'a::{ordered_semidom,recpower}" 

319 
shows "\<lbrakk>a ^ n = b ^ n; 0 \<le> a; 0 \<le> b; 0 < n\<rbrakk> \<Longrightarrow> a = b" 

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320 
by (cases n, simp_all del: power_Suc, rule power_inject_base) 
22955  321 

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322 
text {* The divides relation *} 
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323 

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324 
lemma le_imp_power_dvd: 
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325 
fixes a :: "'a::{comm_semiring_1,recpower}" 
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326 
assumes "m \<le> n" shows "a^m dvd a^n" 
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327 
proof 
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328 
have "a^n = a^(m + (n  m))" 
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329 
using `m \<le> n` by simp 
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330 
also have "\<dots> = a^m * a^(n  m)" 
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331 
by (rule power_add) 
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332 
finally show "a^n = a^m * a^(n  m)" . 
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333 
qed 
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334 

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335 
lemma power_le_dvd: 
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336 
fixes a b :: "'a::{comm_semiring_1,recpower}" 
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337 
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]) 
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30313  341 
lemma dvd_power_same: 
342 
"(x::'a::{comm_semiring_1,recpower}) dvd y \<Longrightarrow> x^n dvd y^n" 

343 
by (induct n) (auto simp add: mult_dvd_mono) 

344 

345 
lemma dvd_power_le: 

346 
"(x::'a::{comm_semiring_1,recpower}) dvd y \<Longrightarrow> m >= n \<Longrightarrow> x^n dvd y^m" 

347 
by(rule power_le_dvd[OF dvd_power_same]) 

348 

349 
lemma dvd_power [simp]: 

350 
"n > 0  (x::'a::{comm_semiring_1,recpower}) = 1 \<Longrightarrow> x dvd x^n" 

351 
apply (erule disjE) 

352 
apply (subgoal_tac "x ^ n = x^(Suc (n  1))") 

353 
apply (erule ssubst) 

354 
apply (subst power_Suc) 

355 
apply auto 

356 
done 

357 

358 

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subsection{*Exponentiation for the Natural Numbers*} 
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25836  361 
instantiation nat :: recpower 
362 
begin 

21456  363 

25836  364 
primrec power_nat where 
365 
"p ^ 0 = (1\<Colon>nat)" 

366 
 "p ^ (Suc n) = (p\<Colon>nat) * (p ^ n)" 

14577  367 

25836  368 
instance proof 
14438  369 
fix z n :: nat 
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show "z^0 = 1" by simp 
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show "z^(Suc n) = z * (z^n)" by simp 
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qed 
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373 

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declare power_nat.simps [simp del] 
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25836  376 
end 
377 

23305  378 
lemma of_nat_power: 
379 
"of_nat (m ^ n) = (of_nat m::'a::{semiring_1,recpower}) ^ n" 

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by (induct n, simp_all add: of_nat_mult) 
23305  381 

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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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384 

25162  385 
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat)  n=0)" 
21413  386 
by (induct "n", auto) 
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30056  388 
lemma nat_power_eq_Suc_0_iff [simp]: 
389 
"((x::nat)^m = Suc 0) = (m = 0  x = Suc 0)" 

390 
by (induct_tac m, auto) 

391 

392 
lemma power_Suc_0[simp]: "(Suc 0)^n = Suc 0" 

393 
by simp 

394 

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text{*Valid for the naturals, but what if @{text"0<i<1"}? 
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Premises cannot be weakened: consider the case where @{term "i=0"}, 
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@{term "m=1"} and @{term "n=0"}.*} 
21413  398 
lemma nat_power_less_imp_less: 
399 
assumes nonneg: "0 < (i\<Colon>nat)" 

400 
assumes less: "i^m < i^n" 

401 
shows "m < n" 

402 
proof (cases "i = 1") 

403 
case True with less power_one [where 'a = nat] show ?thesis by simp 

404 
next 

405 
case False with nonneg have "1 < i" by auto 

406 
from power_strict_increasing_iff [OF this] less show ?thesis .. 

407 
qed 

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408 

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lemma power_diff: 
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410 
assumes nz: "a ~= 0" 
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shows "n <= m ==> (a::'a::{recpower, field}) ^ (mn) = (a^m) / (a^n)" 
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by (induct m n rule: diff_induct) 
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(simp_all add: nonzero_mult_divide_cancel_left nz) 
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414 

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415 

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text{*ML bindings for the general exponentiation theorems*} 
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ML 
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{* 
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val power_0 = thm"power_0"; 
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val power_Suc = thm"power_Suc"; 
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val power_0_Suc = thm"power_0_Suc"; 
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val power_0_left = thm"power_0_left"; 
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val power_one = thm"power_one"; 
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val power_one_right = thm"power_one_right"; 
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val power_add = thm"power_add"; 
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val power_mult = thm"power_mult"; 
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val power_mult_distrib = thm"power_mult_distrib"; 
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val zero_less_power = thm"zero_less_power"; 
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val zero_le_power = thm"zero_le_power"; 
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val one_le_power = thm"one_le_power"; 
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val gt1_imp_ge0 = thm"gt1_imp_ge0"; 
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val power_gt1_lemma = thm"power_gt1_lemma"; 
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val power_gt1 = thm"power_gt1"; 
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val power_le_imp_le_exp = thm"power_le_imp_le_exp"; 
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val power_inject_exp = thm"power_inject_exp"; 
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val power_less_imp_less_exp = thm"power_less_imp_less_exp"; 
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val power_mono = thm"power_mono"; 
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val power_strict_mono = thm"power_strict_mono"; 
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val power_eq_0_iff = thm"power_eq_0_iff"; 
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val field_power_eq_0_iff = thm"power_eq_0_iff"; 
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val field_power_not_zero = thm"field_power_not_zero"; 
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val power_inverse = thm"power_inverse"; 
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val nonzero_power_divide = thm"nonzero_power_divide"; 
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val power_divide = thm"power_divide"; 
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val power_abs = thm"power_abs"; 
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val zero_less_power_abs_iff = thm"zero_less_power_abs_iff"; 
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val zero_le_power_abs = thm "zero_le_power_abs"; 
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val power_minus = thm"power_minus"; 
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val power_Suc_less = thm"power_Suc_less"; 
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val power_strict_decreasing = thm"power_strict_decreasing"; 
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val power_decreasing = thm"power_decreasing"; 
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val power_Suc_less_one = thm"power_Suc_less_one"; 
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val power_increasing = thm"power_increasing"; 
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val power_strict_increasing = thm"power_strict_increasing"; 
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455 
val power_le_imp_le_base = thm"power_le_imp_le_base"; 
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val power_inject_base = thm"power_inject_base"; 
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457 
*} 
14577  458 

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text{*ML bindings for the remaining theorems*} 
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ML 
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461 
{* 
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val nat_one_le_power = thm"nat_one_le_power"; 
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val nat_power_less_imp_less = thm"nat_power_less_imp_less"; 
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val nat_zero_less_power_iff = thm"nat_zero_less_power_iff"; 
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*} 
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466 

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467 
end 