(* Title: HOL/Power.thy ID: $Id: Power.thy,v 1.19 2005/08/26 08:01:06 ballarin Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1997 University of Cambridge *) header{*Exponentiation*} theory Power imports Divides begin subsection{*Powers for Arbitrary Semirings*} axclass recpower ⊆ comm_semiring_1_cancel, power power_0 [simp]: "a ^ 0 = 1" power_Suc: "a ^ (Suc n) = a * (a ^ n)" lemma power_0_Suc [simp]: "(0::'a::recpower) ^ (Suc n) = 0" by (simp add: power_Suc) text{*It looks plausible as a simprule, but its effect can be strange.*} lemma power_0_left: "0^n = (if n=0 then 1 else (0::'a::recpower))" by (induct "n", auto) lemma power_one [simp]: "1^n = (1::'a::recpower)" apply (induct "n") apply (auto simp add: power_Suc) done lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a" by (simp add: power_Suc) lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)" apply (induct "n") apply (simp_all add: power_Suc mult_ac) done lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n" apply (induct "n") apply (simp_all add: power_Suc power_add) done lemma power_mult_distrib: "((a::'a::recpower) * b) ^ n = (a^n) * (b^n)" apply (induct "n") apply (auto simp add: power_Suc mult_ac) done lemma zero_less_power: "0 < (a::'a::{ordered_semidom,recpower}) ==> 0 < a^n" apply (induct "n") apply (simp_all add: power_Suc zero_less_one mult_pos_pos) done lemma zero_le_power: "0 ≤ (a::'a::{ordered_semidom,recpower}) ==> 0 ≤ a^n" apply (simp add: order_le_less) apply (erule disjE) apply (simp_all add: zero_less_power zero_less_one power_0_left) done lemma one_le_power: "1 ≤ (a::'a::{ordered_semidom,recpower}) ==> 1 ≤ a^n" apply (induct "n") apply (simp_all add: power_Suc) apply (rule order_trans [OF _ mult_mono [of 1 _ 1]]) apply (simp_all add: zero_le_one order_trans [OF zero_le_one]) done lemma gt1_imp_ge0: "1 < a ==> 0 ≤ (a::'a::ordered_semidom)" by (simp add: order_trans [OF zero_le_one order_less_imp_le]) lemma power_gt1_lemma: assumes gt1: "1 < (a::'a::{ordered_semidom,recpower})" shows "1 < a * a^n" proof - have "1*1 < a*1" using gt1 by simp also have "… ≤ a * a^n" using gt1 by (simp only: mult_mono gt1_imp_ge0 one_le_power order_less_imp_le zero_le_one order_refl) finally show ?thesis by simp qed lemma power_gt1: "1 < (a::'a::{ordered_semidom,recpower}) ==> 1 < a ^ (Suc n)" by (simp add: power_gt1_lemma power_Suc) lemma power_le_imp_le_exp: assumes gt1: "(1::'a::{recpower,ordered_semidom}) < a" shows "!!n. a^m ≤ a^n ==> m ≤ n" proof (induct m) case 0 show ?case by simp next case (Suc m) show ?case proof (cases n) case 0 from prems have "a * a^m ≤ 1" by (simp add: power_Suc) with gt1 show ?thesis by (force simp only: power_gt1_lemma linorder_not_less [symmetric]) next case (Suc n) from prems show ?thesis by (force dest: mult_left_le_imp_le simp add: power_Suc order_less_trans [OF zero_less_one gt1]) qed qed text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*} lemma power_inject_exp [simp]: "1 < (a::'a::{ordered_semidom,recpower}) ==> (a^m = a^n) = (m=n)" by (force simp add: order_antisym power_le_imp_le_exp) text{*Can relax the first premise to @{term "0<a"} in the case of the natural numbers.*} lemma power_less_imp_less_exp: "[| (1::'a::{recpower,ordered_semidom}) < a; a^m < a^n |] ==> m < n" by (simp add: order_less_le [of m n] order_less_le [of "a^m" "a^n"] power_le_imp_le_exp) lemma power_mono: "[|a ≤ b; (0::'a::{recpower,ordered_semidom}) ≤ a|] ==> a^n ≤ b^n" apply (induct "n") apply (simp_all add: power_Suc) apply (auto intro: mult_mono zero_le_power order_trans [of 0 a b]) done lemma power_strict_mono [rule_format]: "[|a < b; (0::'a::{recpower,ordered_semidom}) ≤ a|] ==> 0 < n --> a^n < b^n" apply (induct "n") apply (auto simp add: mult_strict_mono zero_le_power power_Suc order_le_less_trans [of 0 a b]) done lemma power_eq_0_iff [simp]: "(a^n = 0) = (a = (0::'a::{ordered_idom,recpower}) & 0<n)" apply (induct "n") apply (auto simp add: power_Suc zero_neq_one [THEN not_sym]) done lemma field_power_eq_0_iff [simp]: "(a^n = 0) = (a = (0::'a::{field,recpower}) & 0<n)" apply (induct "n") apply (auto simp add: power_Suc field_mult_eq_0_iff zero_neq_one[THEN not_sym]) done lemma field_power_not_zero: "a ≠ (0::'a::{field,recpower}) ==> a^n ≠ 0" by force lemma nonzero_power_inverse: "a ≠ 0 ==> inverse ((a::'a::{field,recpower}) ^ n) = (inverse a) ^ n" apply (induct "n") apply (auto simp add: power_Suc nonzero_inverse_mult_distrib mult_commute) done text{*Perhaps these should be simprules.*} lemma power_inverse: "inverse ((a::'a::{field,division_by_zero,recpower}) ^ n) = (inverse a) ^ n" apply (induct "n") apply (auto simp add: power_Suc inverse_mult_distrib) done lemma power_one_over: "1 / (a::'a::{field,division_by_zero,recpower})^n = (1 / a)^n" apply (simp add: divide_inverse) apply (rule power_inverse) done lemma nonzero_power_divide: "b ≠ 0 ==> (a/b) ^ n = ((a::'a::{field,recpower}) ^ n) / (b ^ n)" by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse) lemma power_divide: "(a/b) ^ n = ((a::'a::{field,division_by_zero,recpower}) ^ n / b ^ n)" apply (case_tac "b=0", simp add: power_0_left) apply (rule nonzero_power_divide) apply assumption done lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,recpower}) ^ n" apply (induct "n") apply (auto simp add: power_Suc abs_mult) done lemma zero_less_power_abs_iff [simp]: "(0 < (abs a)^n) = (a ≠ (0::'a::{ordered_idom,recpower}) | n=0)" proof (induct "n") case 0 show ?case by (simp add: zero_less_one) next case (Suc n) show ?case by (force simp add: prems power_Suc zero_less_mult_iff) qed lemma zero_le_power_abs [simp]: "(0::'a::{ordered_idom,recpower}) ≤ (abs a)^n" apply (induct "n") apply (auto simp add: zero_le_one zero_le_power) done lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{comm_ring_1,recpower}) ^ n" proof - have "-a = (- 1) * a" by (simp add: minus_mult_left [symmetric]) thus ?thesis by (simp only: power_mult_distrib) qed text{*Lemma for @{text power_strict_decreasing}*} lemma power_Suc_less: "[|(0::'a::{ordered_semidom,recpower}) < a; a < 1|] ==> a * a^n < a^n" apply (induct n) apply (auto simp add: power_Suc mult_strict_left_mono) done lemma power_strict_decreasing: "[|n < N; 0 < a; a < (1::'a::{ordered_semidom,recpower})|] ==> a^N < a^n" apply (erule rev_mp) apply (induct "N") apply (auto simp add: power_Suc power_Suc_less less_Suc_eq) apply (rename_tac m) apply (subgoal_tac "a * a^m < 1 * a^n", simp) apply (rule mult_strict_mono) apply (auto simp add: zero_le_power zero_less_one order_less_imp_le) done text{*Proof resembles that of @{text power_strict_decreasing}*} lemma power_decreasing: "[|n ≤ N; 0 ≤ a; a ≤ (1::'a::{ordered_semidom,recpower})|] ==> a^N ≤ a^n" apply (erule rev_mp) apply (induct "N") apply (auto simp add: power_Suc le_Suc_eq) apply (rename_tac m) apply (subgoal_tac "a * a^m ≤ 1 * a^n", simp) apply (rule mult_mono) apply (auto simp add: zero_le_power zero_le_one) done lemma power_Suc_less_one: "[| 0 < a; a < (1::'a::{ordered_semidom,recpower}) |] ==> a ^ Suc n < 1" apply (insert power_strict_decreasing [of 0 "Suc n" a], simp) done text{*Proof again resembles that of @{text power_strict_decreasing}*} lemma power_increasing: "[|n ≤ N; (1::'a::{ordered_semidom,recpower}) ≤ a|] ==> a^n ≤ a^N" apply (erule rev_mp) apply (induct "N") apply (auto simp add: power_Suc le_Suc_eq) apply (rename_tac m) apply (subgoal_tac "1 * a^n ≤ a * a^m", simp) apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one] zero_le_power) done text{*Lemma for @{text power_strict_increasing}*} lemma power_less_power_Suc: "(1::'a::{ordered_semidom,recpower}) < a ==> a^n < a * a^n" apply (induct n) apply (auto simp add: power_Suc mult_strict_left_mono order_less_trans [OF zero_less_one]) done lemma power_strict_increasing: "[|n < N; (1::'a::{ordered_semidom,recpower}) < a|] ==> a^n < a^N" apply (erule rev_mp) apply (induct "N") apply (auto simp add: power_less_power_Suc power_Suc less_Suc_eq) apply (rename_tac m) apply (subgoal_tac "1 * a^n < a * a^m", simp) apply (rule mult_strict_mono) apply (auto simp add: order_less_trans [OF zero_less_one] zero_le_power order_less_imp_le) done lemma power_increasing_iff [simp]: "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x ≤ b ^ y) = (x ≤ y)" by (blast intro: power_le_imp_le_exp power_increasing order_less_imp_le) lemma power_strict_increasing_iff [simp]: "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x < b ^ y) = (x < y)" by (blast intro: power_less_imp_less_exp power_strict_increasing) lemma power_le_imp_le_base: assumes le: "a ^ Suc n ≤ b ^ Suc n" and xnonneg: "(0::'a::{ordered_semidom,recpower}) ≤ a" and ynonneg: "0 ≤ b" shows "a ≤ b" proof (rule ccontr) assume "~ a ≤ b" then have "b < a" by (simp only: linorder_not_le) then have "b ^ Suc n < a ^ Suc n" by (simp only: prems power_strict_mono) from le and this show "False" by (simp add: linorder_not_less [symmetric]) qed lemma power_inject_base: "[| a ^ Suc n = b ^ Suc n; 0 ≤ a; 0 ≤ b |] ==> a = (b::'a::{ordered_semidom,recpower})" by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym) subsection{*Exponentiation for the Natural Numbers*} primrec (power) "p ^ 0 = 1" "p ^ (Suc n) = (p::nat) * (p ^ n)" instance nat :: recpower proof fix z n :: nat show "z^0 = 1" by simp show "z^(Suc n) = z * (z^n)" by simp qed lemma nat_one_le_power [simp]: "1 ≤ i ==> Suc 0 ≤ i^n" by (insert one_le_power [of i n], simp) lemma le_imp_power_dvd: "!!i::nat. m ≤ n ==> i^m dvd i^n" apply (unfold dvd_def) apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst]) apply (simp add: power_add) done text{*Valid for the naturals, but what if @{text"0<i<1"}? Premises cannot be weakened: consider the case where @{term "i=0"}, @{term "m=1"} and @{term "n=0"}.*} lemma nat_power_less_imp_less: "!!i::nat. [| 0 < i; i^m < i^n |] ==> m < n" apply (rule ccontr) apply (drule leI [THEN le_imp_power_dvd, THEN dvd_imp_le, THEN leD]) apply (erule zero_less_power, auto) done lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x ≠ (0::nat) | n=0)" by (induct "n", auto) lemma power_le_dvd [rule_format]: "k^j dvd n --> i≤j --> k^i dvd (n::nat)" apply (induct "j") apply (simp_all add: le_Suc_eq) apply (blast dest!: dvd_mult_right) done lemma power_dvd_imp_le: "[|i^m dvd i^n; (1::nat) < i|] ==> m ≤ n" apply (rule power_le_imp_le_exp, assumption) apply (erule dvd_imp_le, simp) done lemma power_diff: assumes nz: "a ~= 0" shows "n <= m ==> (a::'a::{recpower, field}) ^ (m-n) = (a^m) / (a^n)" by (induct m n rule: diff_induct) (simp_all add: power_Suc nonzero_mult_divide_cancel_left nz) text{*ML bindings for the general exponentiation theorems*} ML {* val power_0 = thm"power_0"; val power_Suc = thm"power_Suc"; val power_0_Suc = thm"power_0_Suc"; val power_0_left = thm"power_0_left"; val power_one = thm"power_one"; val power_one_right = thm"power_one_right"; val power_add = thm"power_add"; val power_mult = thm"power_mult"; val power_mult_distrib = thm"power_mult_distrib"; val zero_less_power = thm"zero_less_power"; val zero_le_power = thm"zero_le_power"; val one_le_power = thm"one_le_power"; val gt1_imp_ge0 = thm"gt1_imp_ge0"; val power_gt1_lemma = thm"power_gt1_lemma"; val power_gt1 = thm"power_gt1"; val power_le_imp_le_exp = thm"power_le_imp_le_exp"; val power_inject_exp = thm"power_inject_exp"; val power_less_imp_less_exp = thm"power_less_imp_less_exp"; val power_mono = thm"power_mono"; val power_strict_mono = thm"power_strict_mono"; val power_eq_0_iff = thm"power_eq_0_iff"; val field_power_eq_0_iff = thm"field_power_eq_0_iff"; val field_power_not_zero = thm"field_power_not_zero"; val power_inverse = thm"power_inverse"; val nonzero_power_divide = thm"nonzero_power_divide"; val power_divide = thm"power_divide"; val power_abs = thm"power_abs"; val zero_less_power_abs_iff = thm"zero_less_power_abs_iff"; val zero_le_power_abs = thm "zero_le_power_abs"; val power_minus = thm"power_minus"; val power_Suc_less = thm"power_Suc_less"; val power_strict_decreasing = thm"power_strict_decreasing"; val power_decreasing = thm"power_decreasing"; val power_Suc_less_one = thm"power_Suc_less_one"; val power_increasing = thm"power_increasing"; val power_strict_increasing = thm"power_strict_increasing"; val power_le_imp_le_base = thm"power_le_imp_le_base"; val power_inject_base = thm"power_inject_base"; *} text{*ML bindings for the remaining theorems*} ML {* val nat_one_le_power = thm"nat_one_le_power"; val le_imp_power_dvd = thm"le_imp_power_dvd"; val nat_power_less_imp_less = thm"nat_power_less_imp_less"; val nat_zero_less_power_iff = thm"nat_zero_less_power_iff"; val power_le_dvd = thm"power_le_dvd"; val power_dvd_imp_le = thm"power_dvd_imp_le"; *} end
lemma power_0_Suc:
(0::'a) ^ Suc n = (0::'a)
lemma power_0_left:
(0::'a) ^ n = (if n = 0 then 1::'a else 0::'a)
lemma power_one:
(1::'a) ^ n = (1::'a)
lemma power_one_right:
a ^ 1 = a
lemma power_add:
a ^ (m + n) = a ^ m * a ^ n
lemma power_mult:
a ^ (m * n) = (a ^ m) ^ n
lemma power_mult_distrib:
(a * b) ^ n = a ^ n * b ^ n
lemma zero_less_power:
(0::'a) < a ==> (0::'a) < a ^ n
lemma zero_le_power:
(0::'a) ≤ a ==> (0::'a) ≤ a ^ n
lemma one_le_power:
(1::'a) ≤ a ==> (1::'a) ≤ a ^ n
lemma gt1_imp_ge0:
(1::'a) < a ==> (0::'a) ≤ a
lemma power_gt1_lemma:
(1::'a) < a ==> (1::'a) < a * a ^ n
lemma power_gt1:
(1::'a) < a ==> (1::'a) < a ^ Suc n
lemma power_le_imp_le_exp:
[| (1::'a) < a; a ^ m ≤ a ^ n |] ==> m ≤ n
lemma power_inject_exp:
(1::'a) < a ==> (a ^ m = a ^ n) = (m = n)
lemma power_less_imp_less_exp:
[| (1::'a) < a; a ^ m < a ^ n |] ==> m < n
lemma power_mono:
[| a ≤ b; (0::'a) ≤ a |] ==> a ^ n ≤ b ^ n
lemma power_strict_mono:
[| a < b; (0::'a) ≤ a; 0 < n |] ==> a ^ n < b ^ n
lemma power_eq_0_iff:
(a ^ n = (0::'a)) = (a = (0::'a) ∧ 0 < n)
lemma field_power_eq_0_iff:
(a ^ n = (0::'a)) = (a = (0::'a) ∧ 0 < n)
lemma field_power_not_zero:
a ≠ (0::'a) ==> a ^ n ≠ (0::'a)
lemma nonzero_power_inverse:
a ≠ (0::'a) ==> inverse (a ^ n) = inverse a ^ n
lemma power_inverse:
inverse (a ^ n) = inverse a ^ n
lemma power_one_over:
(1::'a) / a ^ n = ((1::'a) / a) ^ n
lemma nonzero_power_divide:
b ≠ (0::'a) ==> (a / b) ^ n = a ^ n / b ^ n
lemma power_divide:
(a / b) ^ n = a ^ n / b ^ n
lemma power_abs:
¦a ^ n¦ = ¦a¦ ^ n
lemma zero_less_power_abs_iff:
((0::'a) < ¦a¦ ^ n) = (a ≠ (0::'a) ∨ n = 0)
lemma zero_le_power_abs:
(0::'a) ≤ ¦a¦ ^ n
lemma power_minus:
(- a) ^ n = (- (1::'a)) ^ n * a ^ n
lemma power_Suc_less:
[| (0::'a) < a; a < (1::'a) |] ==> a * a ^ n < a ^ n
lemma power_strict_decreasing:
[| n < N; (0::'a) < a; a < (1::'a) |] ==> a ^ N < a ^ n
lemma power_decreasing:
[| n ≤ N; (0::'a) ≤ a; a ≤ (1::'a) |] ==> a ^ N ≤ a ^ n
lemma power_Suc_less_one:
[| (0::'a) < a; a < (1::'a) |] ==> a ^ Suc n < (1::'a)
lemma power_increasing:
[| n ≤ N; (1::'a) ≤ a |] ==> a ^ n ≤ a ^ N
lemma power_less_power_Suc:
(1::'a) < a ==> a ^ n < a * a ^ n
lemma power_strict_increasing:
[| n < N; (1::'a) < a |] ==> a ^ n < a ^ N
lemma power_increasing_iff:
(1::'a) < b ==> (b ^ x ≤ b ^ y) = (x ≤ y)
lemma power_strict_increasing_iff:
(1::'a) < b ==> (b ^ x < b ^ y) = (x < y)
lemma power_le_imp_le_base:
[| a ^ Suc n ≤ b ^ Suc n; (0::'a) ≤ a; (0::'a) ≤ b |] ==> a ≤ b
lemma power_inject_base:
[| a ^ Suc n = b ^ Suc n; (0::'a) ≤ a; (0::'a) ≤ b |] ==> a = b
lemma nat_one_le_power:
1 ≤ i ==> Suc 0 ≤ i ^ n
lemma le_imp_power_dvd:
m ≤ n ==> i ^ m dvd i ^ n
lemma nat_power_less_imp_less:
[| 0 < i; i ^ m < i ^ n |] ==> m < n
lemma nat_zero_less_power_iff:
(0 < x ^ n) = (x ≠ 0 ∨ n = 0)
lemma power_le_dvd:
[| k ^ j dvd n; i ≤ j |] ==> k ^ i dvd n
lemma power_dvd_imp_le:
[| i ^ m dvd i ^ n; 1 < i |] ==> m ≤ n
lemma power_diff:
[| a ≠ (0::'a); n ≤ m |] ==> a ^ (m - n) = a ^ m / a ^ n