
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: 
krauss@21199: subsection{*Powers for Arbitrary Monoids*}
paulson@14348: 
haftmann@22390: class recpower = monoid_mult + power +
haftmann@22390:   assumes power_0 [simp]: "a \<^loc>^ 0       = \<^loc>1"
haftmann@22390:   assumes power_Suc:      "a \<^loc>^ Suc n = a \<^loc>* (a \<^loc>^ n)"
paulson@14348: 
krauss@21199: lemma power_0_Suc [simp]: "(0::'a::{recpower,semiring_0}) ^ (Suc n) = 0"
paulson@14348: by (simp add: power_Suc)
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}))"
paulson@15251: by (induct "n", auto)
paulson@14348: 
paulson@15004: lemma power_one [simp]: "1^n = (1::'a::recpower)"
paulson@15251: apply (induct "n")
wenzelm@14577: apply (auto simp add: power_Suc)
paulson@14348: done
paulson@14348: 
paulson@15004: lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a"
paulson@14348: by (simp add: power_Suc)
paulson@14348: 
krauss@21199: lemma power_commutes: "(a::'a::recpower) ^ n * a = a * a ^ n"
krauss@21199: by (induct "n") (simp_all add:power_Suc mult_assoc)
krauss@21199: 
paulson@15004: lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)"
krauss@21199: apply (induct "m")
paulson@14348: apply (simp_all add: power_Suc mult_ac)
paulson@14348: done
paulson@14348: 
paulson@15004: lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n"
krauss@21199: apply (induct "n") 
paulson@14348: apply (simp_all add: power_Suc power_add)
paulson@14348: done
paulson@14348: 
krauss@21199: lemma power_mult_distrib: "((a::'a::{recpower,comm_monoid_mult}) * b) ^ n = (a^n) * (b^n)"
paulson@15251: apply (induct "n")
paulson@14348: apply (auto simp add: power_Suc mult_ac)
paulson@14348: done
paulson@14348: 
paulson@14348: lemma zero_less_power:
paulson@15004:      "0 < (a::'a::{ordered_semidom,recpower}) ==> 0 < a^n"
paulson@15251: apply (induct "n")
avigad@16775: apply (simp_all add: power_Suc zero_less_one mult_pos_pos)
paulson@14348: done
paulson@14348: 
paulson@14348: lemma zero_le_power:
paulson@15004:      "0 \<le> (a::'a::{ordered_semidom,recpower}) ==> 0 \<le> a^n"
paulson@14348: apply (simp add: order_le_less)
wenzelm@14577: apply (erule disjE)
paulson@14348: apply (simp_all add: zero_less_power zero_less_one power_0_left)
paulson@14348: done
paulson@14348: 
paulson@14348: lemma one_le_power:
paulson@15004:      "1 \<le> (a::'a::{ordered_semidom,recpower}) ==> 1 \<le> a^n"
paulson@15251: apply (induct "n")
paulson@14348: apply (simp_all add: power_Suc)
wenzelm@14577: apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
wenzelm@14577: apply (simp_all add: zero_le_one 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: 
paulson@14348: lemma power_gt1:
paulson@15004:      "1 < (a::'a::{ordered_semidom,recpower}) ==> 1 < a ^ (Suc n)"
paulson@14348: by (simp add: power_gt1_lemma power_Suc)
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
wenzelm@14577:     from prems have "a * a^m \<le> 1" by (simp add: power_Suc)
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
wenzelm@14577:           simp add: power_Suc 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")
paulson@14348: apply (simp_all add: power_Suc)
paulson@14348: apply (auto intro: mult_mono zero_le_power 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")
paulson@14348: apply (auto simp add: mult_strict_mono zero_le_power power_Suc
paulson@14348:                       order_le_less_trans [of 0 a b])
paulson@14348: done
paulson@14348: 
paulson@14348: lemma power_eq_0_iff [simp]:
paulson@15004:      "(a^n = 0) = (a = (0::'a::{ordered_idom,recpower}) & 0<n)"
paulson@15251: apply (induct "n")
paulson@14348: apply (auto simp add: power_Suc zero_neq_one [THEN not_sym])
paulson@14348: done
paulson@14348: 
paulson@14348: lemma field_power_eq_0_iff [simp]:
paulson@15004:      "(a^n = 0) = (a = (0::'a::{field,recpower}) & 0<n)"
paulson@15251: apply (induct "n")
paulson@14348: apply (auto simp add: power_Suc field_mult_eq_0_iff zero_neq_one[THEN not_sym])
paulson@14348: done
paulson@14348: 
paulson@15004: lemma field_power_not_zero: "a \<noteq> (0::'a::{field,recpower}) ==> a^n \<noteq> 0"
paulson@14348: by force
paulson@14348: 
paulson@14353: lemma nonzero_power_inverse:
paulson@15004:   "a \<noteq> 0 ==> inverse ((a::'a::{field,recpower}) ^ n) = (inverse a) ^ n"
paulson@15251: apply (induct "n")
paulson@14353: apply (auto simp add: power_Suc nonzero_inverse_mult_distrib mult_commute)
paulson@14353: done
paulson@14353: 
paulson@14348: text{*Perhaps these should be simprules.*}
paulson@14348: lemma power_inverse:
paulson@15004:   "inverse ((a::'a::{field,division_by_zero,recpower}) ^ n) = (inverse a) ^ n"
paulson@15251: apply (induct "n")
paulson@14348: apply (auto simp add: power_Suc inverse_mult_distrib)
paulson@14348: done
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")
paulson@14348: apply (auto simp add: power_Suc abs_mult)
paulson@14348: done
paulson@14348: 
paulson@14353: lemma zero_less_power_abs_iff [simp]:
paulson@15004:      "(0 < (abs a)^n) = (a \<noteq> (0::'a::{ordered_idom,recpower}) | n=0)"
paulson@14353: proof (induct "n")
paulson@14353:   case 0
paulson@14353:     show ?case by (simp add: zero_less_one)
paulson@14353: next
paulson@14353:   case (Suc n)
paulson@14353:     show ?case by (force simp add: prems power_Suc 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"
paulson@15251: apply (induct "n")
paulson@14353: apply (auto simp add: zero_le_one zero_le_power)
paulson@14353: done
paulson@14353: 
paulson@15004: lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{comm_ring_1,recpower}) ^ n"
paulson@14348: proof -
paulson@14348:   have "-a = (- 1) * a"  by (simp add: minus_mult_left [symmetric])
paulson@14348:   thus ?thesis by (simp only: power_mult_distrib)
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)
wenzelm@14577: apply (auto simp add: power_Suc 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")
wenzelm@14577: apply (auto simp add: power_Suc 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)
paulson@14348: apply (auto simp add: zero_le_power zero_less_one 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")
wenzelm@14577: apply (auto simp add: power_Suc  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)
paulson@14348: apply (auto simp add: zero_le_power zero_le_one)
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")
wenzelm@14577: apply (auto simp add: power_Suc 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)
paulson@14348: apply (auto simp add: order_trans [OF zero_le_one] zero_le_power)
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)
wenzelm@14577: apply (auto simp add: power_Suc 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")
wenzelm@14577: apply (auto simp add: power_less_power_Suc 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)
paulson@14348: apply (auto simp add: order_less_trans [OF zero_less_one] zero_le_power
paulson@14348:                  order_less_imp_le)
paulson@14348: done
paulson@14348: 
paulson@15066: lemma power_increasing_iff [simp]: 
paulson@15066:      "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x \<le> b ^ y) = (x \<le> y)"
paulson@15066:   by (blast intro: power_le_imp_le_exp power_increasing order_less_imp_le) 
paulson@15066: 
paulson@15066: lemma power_strict_increasing_iff [simp]:
paulson@15066:      "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x < b ^ y) = (x < y)"
paulson@15066:   by (blast intro: power_less_imp_less_exp power_strict_increasing) 
paulson@15066: 
paulson@14348: lemma power_le_imp_le_base:
paulson@14348:   assumes le: "a ^ Suc n \<le> b ^ Suc n"
paulson@15004:       and xnonneg: "(0::'a::{ordered_semidom,recpower}) \<le> a"
paulson@14348:       and ynonneg: "0 \<le> b"
paulson@14348:   shows "a \<le> b"
paulson@14348:  proof (rule ccontr)
paulson@14348:    assume "~ a \<le> b"
paulson@14348:    then have "b < a" by (simp only: linorder_not_le)
paulson@14348:    then have "b ^ Suc n < a ^ Suc n"
wenzelm@14577:      by (simp only: prems power_strict_mono)
paulson@14348:    from le and this show "False"
paulson@14348:       by (simp add: linorder_not_less [symmetric])
paulson@14348:  qed
wenzelm@14577: 
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: 
paulson@14348: 
paulson@14348: subsection{*Exponentiation for the Natural Numbers*}
paulson@3390: 
haftmann@21456: instance nat :: power ..
haftmann@21456: 
wenzelm@8844: primrec (power)
paulson@3390:   "p ^ 0 = 1"
paulson@3390:   "p ^ (Suc n) = (p::nat) * (p ^ n)"
wenzelm@14577: 
paulson@15004: instance nat :: recpower
paulson@14348: 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: 
paulson@14348: lemma nat_one_le_power [simp]: "1 \<le> i ==> Suc 0 \<le> i^n"
paulson@14348: by (insert one_le_power [of i n], simp)
paulson@14348: 
haftmann@21413: lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat) | n=0)"
haftmann@21413: by (induct "n", auto)
paulson@14348: 
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)
ballarin@17149:     (simp_all add: power_Suc nonzero_mult_divide_cancel_left nz)
ballarin@17149: 
ballarin@17149: 
paulson@14348: text{*ML bindings for the general exponentiation theorems*}
paulson@14348: ML
paulson@14348: {*
paulson@14348: val power_0 = thm"power_0";
paulson@14348: val power_Suc = thm"power_Suc";
paulson@14348: val power_0_Suc = thm"power_0_Suc";
paulson@14348: val power_0_left = thm"power_0_left";
paulson@14348: val power_one = thm"power_one";
paulson@14348: val power_one_right = thm"power_one_right";
paulson@14348: val power_add = thm"power_add";
paulson@14348: val power_mult = thm"power_mult";
paulson@14348: val power_mult_distrib = thm"power_mult_distrib";
paulson@14348: val zero_less_power = thm"zero_less_power";
paulson@14348: val zero_le_power = thm"zero_le_power";
paulson@14348: val one_le_power = thm"one_le_power";
paulson@14348: val gt1_imp_ge0 = thm"gt1_imp_ge0";
paulson@14348: val power_gt1_lemma = thm"power_gt1_lemma";
paulson@14348: val power_gt1 = thm"power_gt1";
paulson@14348: val power_le_imp_le_exp = thm"power_le_imp_le_exp";
paulson@14348: val power_inject_exp = thm"power_inject_exp";
paulson@14348: val power_less_imp_less_exp = thm"power_less_imp_less_exp";
paulson@14348: val power_mono = thm"power_mono";
paulson@14348: val power_strict_mono = thm"power_strict_mono";
paulson@14348: val power_eq_0_iff = thm"power_eq_0_iff";
paulson@14348: val field_power_eq_0_iff = thm"field_power_eq_0_iff";
paulson@14348: val field_power_not_zero = thm"field_power_not_zero";
paulson@14348: val power_inverse = thm"power_inverse";
paulson@14353: val nonzero_power_divide = thm"nonzero_power_divide";
paulson@14353: val power_divide = thm"power_divide";
paulson@14348: val power_abs = thm"power_abs";
paulson@14353: val zero_less_power_abs_iff = thm"zero_less_power_abs_iff";
paulson@14353: val zero_le_power_abs = thm "zero_le_power_abs";
paulson@14348: val power_minus = thm"power_minus";
paulson@14348: val power_Suc_less = thm"power_Suc_less";
paulson@14348: val power_strict_decreasing = thm"power_strict_decreasing";
paulson@14348: val power_decreasing = thm"power_decreasing";
paulson@14348: val power_Suc_less_one = thm"power_Suc_less_one";
paulson@14348: val power_increasing = thm"power_increasing";
paulson@14348: val power_strict_increasing = thm"power_strict_increasing";
paulson@14348: val power_le_imp_le_base = thm"power_le_imp_le_base";
paulson@14348: val power_inject_base = thm"power_inject_base";
paulson@14348: *}
wenzelm@14577: 
paulson@14348: text{*ML bindings for the remaining theorems*}
paulson@14348: ML
paulson@14348: {*
paulson@14348: val nat_one_le_power = thm"nat_one_le_power";
paulson@14348: val nat_power_less_imp_less = thm"nat_power_less_imp_less";
paulson@14348: val nat_zero_less_power_iff = thm"nat_zero_less_power_iff";
paulson@14348: *}
paulson@3390: 
paulson@3390: end
paulson@3390: 




