
paulson@3390: (*  Title:      HOL/Power.thy
paulson@3390:     ID:         $Id$
paulson@3390:     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
paulson@3390:     Copyright   1997  University of Cambridge
paulson@3390: 
paulson@3390: *)
paulson@3390: 
nipkow@16733: header{*Exponentiation*}
paulson@14348: 
nipkow@15131: theory Power
haftmann@21413: imports Nat
nipkow@15131: begin
paulson@14348: 
haftmann@29608: class power =
haftmann@25062:   fixes power :: "'a \<Rightarrow> nat \<Rightarrow> 'a"            (infixr "^" 80)
haftmann@24996: 
krauss@21199: subsection{*Powers for Arbitrary Monoids*}
paulson@14348: 
haftmann@22390: class recpower = monoid_mult + power +
haftmann@25062:   assumes power_0 [simp]: "a ^ 0       = 1"
huffman@30273:   assumes power_Suc [simp]: "a ^ Suc n = a * (a ^ n)"
paulson@14348: 
krauss@21199: lemma power_0_Suc [simp]: "(0::'a::{recpower,semiring_0}) ^ (Suc n) = 0"
huffman@30273:   by simp
paulson@14348: 
paulson@14348: text{*It looks plausible as a simprule, but its effect can be strange.*}
krauss@21199: lemma power_0_left: "0^n = (if n=0 then 1 else (0::'a::{recpower,semiring_0}))"
haftmann@23183:   by (induct n) simp_all
paulson@14348: 
paulson@15004: lemma power_one [simp]: "1^n = (1::'a::recpower)"
huffman@30273:   by (induct n) simp_all
paulson@14348: 
paulson@15004: lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a"
huffman@30273:   unfolding One_nat_def by simp
paulson@14348: 
krauss@21199: lemma power_commutes: "(a::'a::recpower) ^ n * a = a * a ^ n"
huffman@30273:   by (induct n) (simp_all add: mult_assoc)
krauss@21199: 
huffman@28131: lemma power_Suc2: "(a::'a::recpower) ^ Suc n = a ^ n * a"
huffman@30273:   by (simp add: power_commutes)
huffman@28131: 
paulson@15004: lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)"
huffman@30273:   by (induct m) (simp_all add: mult_ac)
paulson@14348: 
paulson@15004: lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n"
huffman@30273:   by (induct n) (simp_all add: power_add)
paulson@14348: 
krauss@21199: lemma power_mult_distrib: "((a::'a::{recpower,comm_monoid_mult}) * b) ^ n = (a^n) * (b^n)"
huffman@30273:   by (induct n) (simp_all add: mult_ac)
paulson@14348: 
nipkow@25874: lemma zero_less_power[simp]:
paulson@15004:      "0 < (a::'a::{ordered_semidom,recpower}) ==> 0 < a^n"
huffman@30273: by (induct n) (simp_all add: mult_pos_pos)
paulson@14348: 
nipkow@25874: lemma zero_le_power[simp]:
paulson@15004:      "0 \<le> (a::'a::{ordered_semidom,recpower}) ==> 0 \<le> a^n"
huffman@30273: by (induct n) (simp_all add: mult_nonneg_nonneg)
paulson@14348: 
nipkow@25874: lemma one_le_power[simp]:
paulson@15004:      "1 \<le> (a::'a::{ordered_semidom,recpower}) ==> 1 \<le> a^n"
paulson@15251: apply (induct "n")
huffman@30273: apply simp_all
wenzelm@14577: apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
huffman@30273: apply (simp_all add: order_trans [OF zero_le_one])
paulson@14348: done
paulson@14348: 
obua@14738: lemma gt1_imp_ge0: "1 < a ==> 0 \<le> (a::'a::ordered_semidom)"
paulson@14348:   by (simp add: order_trans [OF zero_le_one order_less_imp_le])
paulson@14348: 
paulson@14348: lemma power_gt1_lemma:
paulson@15004:   assumes gt1: "1 < (a::'a::{ordered_semidom,recpower})"
wenzelm@14577:   shows "1 < a * a^n"
paulson@14348: proof -
wenzelm@14577:   have "1*1 < a*1" using gt1 by simp
wenzelm@14577:   also have "\<dots> \<le> a * a^n" using gt1
wenzelm@14577:     by (simp only: mult_mono gt1_imp_ge0 one_le_power order_less_imp_le
wenzelm@14577:         zero_le_one order_refl)
wenzelm@14577:   finally show ?thesis by simp
paulson@14348: qed
paulson@14348: 
nipkow@25874: lemma one_less_power[simp]:
huffman@24376:   "\<lbrakk>1 < (a::'a::{ordered_semidom,recpower}); 0 < n\<rbrakk> \<Longrightarrow> 1 < a ^ n"
huffman@30273: by (cases n, simp_all add: power_gt1_lemma)
huffman@24376: 
paulson@14348: lemma power_gt1:
paulson@15004:      "1 < (a::'a::{ordered_semidom,recpower}) ==> 1 < a ^ (Suc n)"
huffman@30273: by (simp add: power_gt1_lemma)
paulson@14348: 
paulson@14348: lemma power_le_imp_le_exp:
paulson@15004:   assumes gt1: "(1::'a::{recpower,ordered_semidom}) < a"
wenzelm@14577:   shows "!!n. a^m \<le> a^n ==> m \<le> n"
wenzelm@14577: proof (induct m)
paulson@14348:   case 0
wenzelm@14577:   show ?case by simp
paulson@14348: next
paulson@14348:   case (Suc m)
wenzelm@14577:   show ?case
wenzelm@14577:   proof (cases n)
wenzelm@14577:     case 0
huffman@30273:     from prems have "a * a^m \<le> 1" by simp
wenzelm@14577:     with gt1 show ?thesis
wenzelm@14577:       by (force simp only: power_gt1_lemma
wenzelm@14577:           linorder_not_less [symmetric])
wenzelm@14577:   next
wenzelm@14577:     case (Suc n)
wenzelm@14577:     from prems show ?thesis
wenzelm@14577:       by (force dest: mult_left_le_imp_le
huffman@30273:           simp add: order_less_trans [OF zero_less_one gt1])
wenzelm@14577:   qed
paulson@14348: qed
paulson@14348: 
wenzelm@14577: text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
paulson@14348: lemma power_inject_exp [simp]:
paulson@15004:      "1 < (a::'a::{ordered_semidom,recpower}) ==> (a^m = a^n) = (m=n)"
wenzelm@14577:   by (force simp add: order_antisym power_le_imp_le_exp)
paulson@14348: 
paulson@14348: text{*Can relax the first premise to @{term "0<a"} in the case of the
paulson@14348: natural numbers.*}
paulson@14348: lemma power_less_imp_less_exp:
paulson@15004:      "[| (1::'a::{recpower,ordered_semidom}) < a; a^m < a^n |] ==> m < n"
wenzelm@14577: by (simp add: order_less_le [of m n] order_less_le [of "a^m" "a^n"]
wenzelm@14577:               power_le_imp_le_exp)
paulson@14348: 
paulson@14348: 
paulson@14348: lemma power_mono:
paulson@15004:      "[|a \<le> b; (0::'a::{recpower,ordered_semidom}) \<le> a|] ==> a^n \<le> b^n"
paulson@15251: apply (induct "n")
huffman@30273: apply simp_all
nipkow@25874: apply (auto intro: mult_mono order_trans [of 0 a b])
paulson@14348: done
paulson@14348: 
paulson@14348: lemma power_strict_mono [rule_format]:
paulson@15004:      "[|a < b; (0::'a::{recpower,ordered_semidom}) \<le> a|]
wenzelm@14577:       ==> 0 < n --> a^n < b^n"
paulson@15251: apply (induct "n")
huffman@30273: apply (auto simp add: mult_strict_mono order_le_less_trans [of 0 a b])
paulson@14348: done
paulson@14348: 
paulson@14348: lemma power_eq_0_iff [simp]:
nipkow@30056:   "(a^n = 0) \<longleftrightarrow>
nipkow@30056:    (a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,recpower}) & n\<noteq>0)"
paulson@15251: apply (induct "n")
huffman@30273: apply (auto simp add: no_zero_divisors)
paulson@14348: done
paulson@14348: 
nipkow@30056: 
nipkow@25134: lemma field_power_not_zero:
nipkow@25134:   "a \<noteq> (0::'a::{ring_1_no_zero_divisors,recpower}) ==> a^n \<noteq> 0"
paulson@14348: by force
paulson@14348: 
paulson@14353: lemma nonzero_power_inverse:
huffman@22991:   fixes a :: "'a::{division_ring,recpower}"
huffman@22991:   shows "a \<noteq> 0 ==> inverse (a ^ n) = (inverse a) ^ n"
paulson@15251: apply (induct "n")
huffman@30273: apply (auto simp add: nonzero_inverse_mult_distrib power_commutes)
huffman@22991: done (* TODO: reorient or rename to nonzero_inverse_power *)
paulson@14353: 
paulson@14348: text{*Perhaps these should be simprules.*}
paulson@14348: lemma power_inverse:
huffman@22991:   fixes a :: "'a::{division_ring,division_by_zero,recpower}"
huffman@22991:   shows "inverse (a ^ n) = (inverse a) ^ n"
huffman@22991: apply (cases "a = 0")
huffman@22991: apply (simp add: power_0_left)
huffman@22991: apply (simp add: nonzero_power_inverse)
huffman@22991: done (* TODO: reorient or rename to inverse_power *)
paulson@14348: 
avigad@16775: lemma power_one_over: "1 / (a::'a::{field,division_by_zero,recpower})^n = 
avigad@16775:     (1 / a)^n"
avigad@16775: apply (simp add: divide_inverse)
avigad@16775: apply (rule power_inverse)
avigad@16775: done
avigad@16775: 
wenzelm@14577: lemma nonzero_power_divide:
paulson@15004:     "b \<noteq> 0 ==> (a/b) ^ n = ((a::'a::{field,recpower}) ^ n) / (b ^ n)"
paulson@14353: by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
paulson@14353: 
wenzelm@14577: lemma power_divide:
paulson@15004:     "(a/b) ^ n = ((a::'a::{field,division_by_zero,recpower}) ^ n / b ^ n)"
paulson@14353: apply (case_tac "b=0", simp add: power_0_left)
wenzelm@14577: apply (rule nonzero_power_divide)
wenzelm@14577: apply assumption
paulson@14353: done
paulson@14353: 
paulson@15004: lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,recpower}) ^ n"
paulson@15251: apply (induct "n")
huffman@30273: apply (auto simp add: abs_mult)
paulson@14348: done
paulson@14348: 
paulson@30730: lemma abs_power_minus [simp]:
paulson@30730:   fixes a:: "'a::{ordered_idom,recpower}" shows "abs((-a) ^ n) = abs(a ^ n)"
paulson@30730:   by (simp add: abs_minus_cancel power_abs) 
paulson@30730: 
paulson@24286: lemma zero_less_power_abs_iff [simp,noatp]:
paulson@15004:      "(0 < (abs a)^n) = (a \<noteq> (0::'a::{ordered_idom,recpower}) | n=0)"
paulson@14353: proof (induct "n")
paulson@14353:   case 0
huffman@30273:     show ?case by simp
paulson@14353: next
paulson@14353:   case (Suc n)
huffman@30273:     show ?case by (auto simp add: prems zero_less_mult_iff)
paulson@14353: qed
paulson@14353: 
paulson@14353: lemma zero_le_power_abs [simp]:
paulson@15004:      "(0::'a::{ordered_idom,recpower}) \<le> (abs a)^n"
huffman@22957: by (rule zero_le_power [OF abs_ge_zero])
paulson@14353: 
huffman@28131: lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{ring_1,recpower}) ^ n"
huffman@28131: proof (induct n)
huffman@28131:   case 0 show ?case by simp
huffman@28131: next
huffman@28131:   case (Suc n) then show ?case
huffman@30273:     by (simp del: power_Suc add: power_Suc2 mult_assoc)
paulson@14348: qed
paulson@14348: 
paulson@14348: text{*Lemma for @{text power_strict_decreasing}*}
paulson@14348: lemma power_Suc_less:
paulson@15004:      "[|(0::'a::{ordered_semidom,recpower}) < a; a < 1|]
paulson@14348:       ==> a * a^n < a^n"
paulson@15251: apply (induct n)
huffman@30273: apply (auto simp add: mult_strict_left_mono)
paulson@14348: done
paulson@14348: 
paulson@14348: lemma power_strict_decreasing:
paulson@15004:      "[|n < N; 0 < a; a < (1::'a::{ordered_semidom,recpower})|]
paulson@14348:       ==> a^N < a^n"
wenzelm@14577: apply (erule rev_mp)
paulson@15251: apply (induct "N")
huffman@30273: apply (auto simp add: power_Suc_less less_Suc_eq)
wenzelm@14577: apply (rename_tac m)
paulson@14348: apply (subgoal_tac "a * a^m < 1 * a^n", simp)
wenzelm@14577: apply (rule mult_strict_mono)
huffman@30273: apply (auto simp add: order_less_imp_le)
paulson@14348: done
paulson@14348: 
paulson@14348: text{*Proof resembles that of @{text power_strict_decreasing}*}
paulson@14348: lemma power_decreasing:
paulson@15004:      "[|n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semidom,recpower})|]
paulson@14348:       ==> a^N \<le> a^n"
wenzelm@14577: apply (erule rev_mp)
paulson@15251: apply (induct "N")
huffman@30273: apply (auto simp add: le_Suc_eq)
wenzelm@14577: apply (rename_tac m)
paulson@14348: apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp)
wenzelm@14577: apply (rule mult_mono)
huffman@30273: apply auto
paulson@14348: done
paulson@14348: 
paulson@14348: lemma power_Suc_less_one:
paulson@15004:      "[| 0 < a; a < (1::'a::{ordered_semidom,recpower}) |] ==> a ^ Suc n < 1"
wenzelm@14577: apply (insert power_strict_decreasing [of 0 "Suc n" a], simp)
paulson@14348: done
paulson@14348: 
paulson@14348: text{*Proof again resembles that of @{text power_strict_decreasing}*}
paulson@14348: lemma power_increasing:
paulson@15004:      "[|n \<le> N; (1::'a::{ordered_semidom,recpower}) \<le> a|] ==> a^n \<le> a^N"
wenzelm@14577: apply (erule rev_mp)
paulson@15251: apply (induct "N")
huffman@30273: apply (auto simp add: le_Suc_eq)
paulson@14348: apply (rename_tac m)
paulson@14348: apply (subgoal_tac "1 * a^n \<le> a * a^m", simp)
wenzelm@14577: apply (rule mult_mono)
nipkow@25874: apply (auto simp add: order_trans [OF zero_le_one])
paulson@14348: done
paulson@14348: 
paulson@14348: text{*Lemma for @{text power_strict_increasing}*}
paulson@14348: lemma power_less_power_Suc:
paulson@15004:      "(1::'a::{ordered_semidom,recpower}) < a ==> a^n < a * a^n"
paulson@15251: apply (induct n)
huffman@30273: apply (auto simp add: mult_strict_left_mono order_less_trans [OF zero_less_one])
paulson@14348: done
paulson@14348: 
paulson@14348: lemma power_strict_increasing:
paulson@15004:      "[|n < N; (1::'a::{ordered_semidom,recpower}) < a|] ==> a^n < a^N"
wenzelm@14577: apply (erule rev_mp)
paulson@15251: apply (induct "N")
huffman@30273: apply (auto simp add: power_less_power_Suc less_Suc_eq)
paulson@14348: apply (rename_tac m)
paulson@14348: apply (subgoal_tac "1 * a^n < a * a^m", simp)
wenzelm@14577: apply (rule mult_strict_mono)
nipkow@25874: apply (auto simp add: order_less_trans [OF zero_less_one] order_less_imp_le)
paulson@14348: done
paulson@14348: 
nipkow@25134: lemma power_increasing_iff [simp]:
nipkow@25134:   "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x \<le> b ^ y) = (x \<le> y)"
nipkow@25134: by (blast intro: power_le_imp_le_exp power_increasing order_less_imp_le) 
paulson@15066: 
paulson@15066: lemma power_strict_increasing_iff [simp]:
nipkow@25134:   "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x < b ^ y) = (x < y)"
nipkow@25134: by (blast intro: power_less_imp_less_exp power_strict_increasing) 
paulson@15066: 
paulson@14348: lemma power_le_imp_le_base:
nipkow@25134: assumes le: "a ^ Suc n \<le> b ^ Suc n"
nipkow@25134:     and ynonneg: "(0::'a::{ordered_semidom,recpower}) \<le> b"
nipkow@25134: shows "a \<le> b"
nipkow@25134: proof (rule ccontr)
nipkow@25134:   assume "~ a \<le> b"
nipkow@25134:   then have "b < a" by (simp only: linorder_not_le)
nipkow@25134:   then have "b ^ Suc n < a ^ Suc n"
nipkow@25134:     by (simp only: prems power_strict_mono)
nipkow@25134:   from le and this show "False"
nipkow@25134:     by (simp add: linorder_not_less [symmetric])
nipkow@25134: qed
wenzelm@14577: 
huffman@22853: lemma power_less_imp_less_base:
huffman@22853:   fixes a b :: "'a::{ordered_semidom,recpower}"
huffman@22853:   assumes less: "a ^ n < b ^ n"
huffman@22853:   assumes nonneg: "0 \<le> b"
huffman@22853:   shows "a < b"
huffman@22853: proof (rule contrapos_pp [OF less])
huffman@22853:   assume "~ a < b"
huffman@22853:   hence "b \<le> a" by (simp only: linorder_not_less)
huffman@22853:   hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
huffman@22853:   thus "~ a ^ n < b ^ n" by (simp only: linorder_not_less)
huffman@22853: qed
huffman@22853: 
paulson@14348: lemma power_inject_base:
wenzelm@14577:      "[| a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b |]
paulson@15004:       ==> a = (b::'a::{ordered_semidom,recpower})"
paulson@14348: by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym)
paulson@14348: 
huffman@22955: lemma power_eq_imp_eq_base:
huffman@22955:   fixes a b :: "'a::{ordered_semidom,recpower}"
huffman@22955:   shows "\<lbrakk>a ^ n = b ^ n; 0 \<le> a; 0 \<le> b; 0 < n\<rbrakk> \<Longrightarrow> a = b"
huffman@30273: by (cases n, simp_all del: power_Suc, rule power_inject_base)
huffman@22955: 
huffman@29978: text {* The divides relation *}
huffman@29978: 
huffman@29978: lemma le_imp_power_dvd:
huffman@29978:   fixes a :: "'a::{comm_semiring_1,recpower}"
huffman@29978:   assumes "m \<le> n" shows "a^m dvd a^n"
huffman@29978: proof
huffman@29978:   have "a^n = a^(m + (n - m))"
huffman@29978:     using `m \<le> n` by simp
huffman@29978:   also have "\<dots> = a^m * a^(n - m)"
huffman@29978:     by (rule power_add)
huffman@29978:   finally show "a^n = a^m * a^(n - m)" .
huffman@29978: qed
huffman@29978: 
huffman@29978: lemma power_le_dvd:
huffman@29978:   fixes a b :: "'a::{comm_semiring_1,recpower}"
huffman@29978:   shows "a^n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a^m dvd b"
huffman@29978:   by (rule dvd_trans [OF le_imp_power_dvd])
huffman@29978: 
paulson@14348: 
nipkow@30313: lemma dvd_power_same:
nipkow@30313:   "(x::'a::{comm_semiring_1,recpower}) dvd y \<Longrightarrow> x^n dvd y^n"
nipkow@30313: by (induct n) (auto simp add: mult_dvd_mono)
nipkow@30313: 
nipkow@30313: lemma dvd_power_le:
nipkow@30313:   "(x::'a::{comm_semiring_1,recpower}) dvd y \<Longrightarrow> m >= n \<Longrightarrow> x^n dvd y^m"
nipkow@30313: by(rule power_le_dvd[OF dvd_power_same])
nipkow@30313: 
nipkow@30313: lemma dvd_power [simp]:
nipkow@30313:   "n > 0 | (x::'a::{comm_semiring_1,recpower}) = 1 \<Longrightarrow> x dvd x^n"
nipkow@30313: apply (erule disjE)
nipkow@30313:  apply (subgoal_tac "x ^ n = x^(Suc (n - 1))")
nipkow@30313:   apply (erule ssubst)
nipkow@30313:   apply (subst power_Suc)
nipkow@30313:   apply auto
nipkow@30313: done
nipkow@30313: 
nipkow@30313: 
paulson@14348: subsection{*Exponentiation for the Natural Numbers*}
paulson@3390: 
haftmann@25836: instantiation nat :: recpower
haftmann@25836: begin
haftmann@21456: 
haftmann@25836: primrec power_nat where
haftmann@25836:   "p ^ 0 = (1\<Colon>nat)"
haftmann@25836:   | "p ^ (Suc n) = (p\<Colon>nat) * (p ^ n)"
wenzelm@14577: 
haftmann@25836: instance proof
paulson@14438:   fix z n :: nat
paulson@14348:   show "z^0 = 1" by simp
paulson@14348:   show "z^(Suc n) = z * (z^n)" by simp
paulson@14348: qed
paulson@14348: 
huffman@30273: declare power_nat.simps [simp del]
huffman@30273: 
haftmann@25836: end
haftmann@25836: 
huffman@23305: lemma of_nat_power:
huffman@23305:   "of_nat (m ^ n) = (of_nat m::'a::{semiring_1,recpower}) ^ n"
huffman@30273: by (induct n, simp_all add: of_nat_mult)
huffman@23305: 
huffman@30079: lemma nat_one_le_power [simp]: "Suc 0 \<le> i ==> Suc 0 \<le> i^n"
huffman@30079: by (rule one_le_power [of i n, unfolded One_nat_def])
paulson@14348: 
nipkow@25162: lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
haftmann@21413: by (induct "n", auto)
paulson@14348: 
nipkow@30056: lemma nat_power_eq_Suc_0_iff [simp]: 
nipkow@30056:   "((x::nat)^m = Suc 0) = (m = 0 | x = Suc 0)"
nipkow@30056: by (induct_tac m, auto)
nipkow@30056: 
nipkow@30056: lemma power_Suc_0[simp]: "(Suc 0)^n = Suc 0"
nipkow@30056: by simp
nipkow@30056: 
paulson@14348: text{*Valid for the naturals, but what if @{text"0<i<1"}?
paulson@14348: Premises cannot be weakened: consider the case where @{term "i=0"},
paulson@14348: @{term "m=1"} and @{term "n=0"}.*}
haftmann@21413: lemma nat_power_less_imp_less:
haftmann@21413:   assumes nonneg: "0 < (i\<Colon>nat)"
haftmann@21413:   assumes less: "i^m < i^n"
haftmann@21413:   shows "m < n"
haftmann@21413: proof (cases "i = 1")
haftmann@21413:   case True with less power_one [where 'a = nat] show ?thesis by simp
haftmann@21413: next
haftmann@21413:   case False with nonneg have "1 < i" by auto
haftmann@21413:   from power_strict_increasing_iff [OF this] less show ?thesis ..
haftmann@21413: qed
paulson@14348: 
ballarin@17149: lemma power_diff:
ballarin@17149:   assumes nz: "a ~= 0"
ballarin@17149:   shows "n <= m ==> (a::'a::{recpower, field}) ^ (m-n) = (a^m) / (a^n)"
ballarin@17149:   by (induct m n rule: diff_induct)
huffman@30273:     (simp_all add: nonzero_mult_divide_cancel_left nz)
ballarin@17149: 
paulson@3390: end




