(* Title: HOL/Power.thy ID: $Id$ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1997 University of Cambridge *) header{*Exponentiation*} theory Power imports Nat begin class power = fixes power :: "'a \ nat \ 'a" (infixr "^" 80) subsection{*Powers for Arbitrary Monoids*} class recpower = monoid_mult + power + assumes power_0 [simp]: "a ^ 0 = 1" assumes power_Suc [simp]: "a ^ Suc n = a * (a ^ n)" lemma power_0_Suc [simp]: "(0::'a::{recpower,semiring_0}) ^ (Suc n) = 0" by simp 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,semiring_0}))" by (induct n) simp_all lemma power_one [simp]: "1^n = (1::'a::recpower)" by (induct n) simp_all lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a" unfolding One_nat_def by simp lemma power_commutes: "(a::'a::recpower) ^ n * a = a * a ^ n" by (induct n) (simp_all add: mult_assoc) lemma power_Suc2: "(a::'a::recpower) ^ Suc n = a ^ n * a" by (simp add: power_commutes) lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)" by (induct m) (simp_all add: mult_ac) lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n" by (induct n) (simp_all add: power_add) lemma power_mult_distrib: "((a::'a::{recpower,comm_monoid_mult}) * b) ^ n = (a^n) * (b^n)" by (induct n) (simp_all add: mult_ac) lemma zero_less_power[simp]: "0 < (a::'a::{ordered_semidom,recpower}) ==> 0 < a^n" by (induct n) (simp_all add: mult_pos_pos) lemma zero_le_power[simp]: "0 \ (a::'a::{ordered_semidom,recpower}) ==> 0 \ a^n" by (induct n) (simp_all add: mult_nonneg_nonneg) lemma one_le_power[simp]: "1 \ (a::'a::{ordered_semidom,recpower}) ==> 1 \ a^n" apply (induct "n") apply simp_all apply (rule order_trans [OF _ mult_mono [of 1 _ 1]]) apply (simp_all add: 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 one_less_power[simp]: "\1 < (a::'a::{ordered_semidom,recpower}); 0 < n\ \ 1 < a ^ n" by (cases n, simp_all add: power_gt1_lemma) lemma power_gt1: "1 < (a::'a::{ordered_semidom,recpower}) ==> 1 < a ^ (Suc n)" by (simp add: power_gt1_lemma) 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 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: order_less_trans [OF zero_less_one gt1]) qed qed text{*Surely we can strengthen this? It holds for @{text "0 (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 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 apply (auto intro: mult_mono 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 order_le_less_trans [of 0 a b]) done lemma power_eq_0_iff [simp]: "(a^n = 0) \ (a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,recpower}) & n\0)" apply (induct "n") apply (auto simp add: no_zero_divisors) done lemma field_power_not_zero: "a \ (0::'a::{ring_1_no_zero_divisors,recpower}) ==> a^n \ 0" by force lemma nonzero_power_inverse: fixes a :: "'a::{division_ring,recpower}" shows "a \ 0 ==> inverse (a ^ n) = (inverse a) ^ n" apply (induct "n") apply (auto simp add: nonzero_inverse_mult_distrib power_commutes) done (* TODO: reorient or rename to nonzero_inverse_power *) text{*Perhaps these should be simprules.*} lemma power_inverse: fixes a :: "'a::{division_ring,division_by_zero,recpower}" shows "inverse (a ^ n) = (inverse a) ^ n" apply (cases "a = 0") apply (simp add: power_0_left) apply (simp add: nonzero_power_inverse) done (* TODO: reorient or rename to inverse_power *) 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: abs_mult) done lemma zero_less_power_abs_iff [simp,noatp]: "(0 < (abs a)^n) = (a \ (0::'a::{ordered_idom,recpower}) | n=0)" proof (induct "n") case 0 show ?case by simp next case (Suc n) show ?case by (auto simp add: prems zero_less_mult_iff) qed lemma zero_le_power_abs [simp]: "(0::'a::{ordered_idom,recpower}) \ (abs a)^n" by (rule zero_le_power [OF abs_ge_zero]) lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{ring_1,recpower}) ^ n" proof (induct n) case 0 show ?case by simp next case (Suc n) then show ?case by (simp del: power_Suc add: power_Suc2 mult_assoc) 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: 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_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: 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: le_Suc_eq) apply (rename_tac m) apply (subgoal_tac "a * a^m \ 1 * a^n", simp) apply (rule mult_mono) apply auto 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: 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]) 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: 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 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] 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 ynonneg: "(0::'a::{ordered_semidom,recpower}) \ 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_less_imp_less_base: fixes a b :: "'a::{ordered_semidom,recpower}" assumes less: "a ^ n < b ^ n" assumes nonneg: "0 \ b" shows "a < b" proof (rule contrapos_pp [OF less]) assume "~ a < b" hence "b \ a" by (simp only: linorder_not_less) hence "b ^ n \ a ^ n" using nonneg by (rule power_mono) thus "~ a ^ n < b ^ n" by (simp only: linorder_not_less) 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) lemma power_eq_imp_eq_base: fixes a b :: "'a::{ordered_semidom,recpower}" shows "\a ^ n = b ^ n; 0 \ a; 0 \ b; 0 < n\ \ a = b" by (cases n, simp_all del: power_Suc, rule power_inject_base) text {* The divides relation *} lemma le_imp_power_dvd: fixes a :: "'a::{comm_semiring_1,recpower}" assumes "m \ n" shows "a^m dvd a^n" proof have "a^n = a^(m + (n - m))" using `m \ n` by simp also have "\ = a^m * a^(n - m)" by (rule power_add) finally show "a^n = a^m * a^(n - m)" . qed lemma power_le_dvd: fixes a b :: "'a::{comm_semiring_1,recpower}" shows "a^n dvd b \ m \ n \ a^m dvd b" by (rule dvd_trans [OF le_imp_power_dvd]) subsection{*Exponentiation for the Natural Numbers*} instantiation nat :: recpower begin primrec power_nat where "p ^ 0 = (1\nat)" | "p ^ (Suc n) = (p\nat) * (p ^ n)" instance proof fix z n :: nat show "z^0 = 1" by simp show "z^(Suc n) = z * (z^n)" by simp qed declare power_nat.simps [simp del] end lemma of_nat_power: "of_nat (m ^ n) = (of_nat m::'a::{semiring_1,recpower}) ^ n" by (induct n, simp_all add: of_nat_mult) lemma nat_one_le_power [simp]: "Suc 0 \ i ==> Suc 0 \ i^n" by (rule one_le_power [of i n, unfolded One_nat_def]) lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)" by (induct "n", auto) lemma nat_power_eq_Suc_0_iff [simp]: "((x::nat)^m = Suc 0) = (m = 0 | x = Suc 0)" by (induct_tac m, auto) lemma power_Suc_0[simp]: "(Suc 0)^n = Suc 0" by simp text{*Valid for the naturals, but what if @{text"0nat)" assumes less: "i^m < i^n" shows "m < n" proof (cases "i = 1") case True with less power_one [where 'a = nat] show ?thesis by simp next case False with nonneg have "1 < i" by auto from power_strict_increasing_iff [OF this] less show ?thesis .. qed 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: 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"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 nat_power_less_imp_less = thm"nat_power_less_imp_less"; val nat_zero_less_power_iff = thm"nat_zero_less_power_iff"; *} end