(* Title: HOL/Power.thy ID: $Id$ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1997 University of Cambridge *) header{*Exponentiation and Binomial Coefficients*} 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) 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^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 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 (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 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 not_less_iff_le [THEN iffD2, THEN add_diff_inverse, THEN subst]) apply (simp add: power_add) done text{*Valid for the naturals, but what if @{text"0 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 subsection{*Binomial Coefficients*} text{*This development is based on the work of Andy Gordon and Florian Kammueller*} consts binomial :: "[nat,nat] => nat" (infixl "choose" 65) primrec binomial_0: "(0 choose k) = (if k = 0 then 1 else 0)" binomial_Suc: "(Suc n choose k) = (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))" lemma binomial_n_0 [simp]: "(n choose 0) = 1" by (case_tac "n", simp_all) lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0" by simp lemma binomial_Suc_Suc [simp]: "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)" by simp lemma binomial_eq_0 [rule_format]: "\k. n < k --> (n choose k) = 0" apply (induct "n", auto) apply (erule allE) apply (erule mp, arith) done declare binomial_0 [simp del] binomial_Suc [simp del] lemma binomial_n_n [simp]: "(n choose n) = 1" apply (induct "n") apply (simp_all add: binomial_eq_0) done lemma binomial_Suc_n [simp]: "(Suc n choose n) = Suc n" by (induct "n", simp_all) lemma binomial_1 [simp]: "(n choose Suc 0) = n" by (induct "n", simp_all) lemma zero_less_binomial [rule_format]: "k \ n --> 0 < (n choose k)" by (rule_tac m = n and n = k in diff_induct, simp_all) lemma binomial_eq_0_iff: "(n choose k = 0) = (nn)" by (simp add: linorder_not_less [symmetric] binomial_eq_0_iff [symmetric]) (*Might be more useful if re-oriented*) lemma Suc_times_binomial_eq [rule_format]: "\k. k \ n --> Suc n * (n choose k) = (Suc n choose Suc k) * Suc k" apply (induct "n") apply (simp add: binomial_0, clarify) apply (case_tac "k") apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq binomial_eq_0) done text{*This is the well-known version, but it's harder to use because of the need to reason about division.*} lemma binomial_Suc_Suc_eq_times: "k \ n ==> (Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k" by (simp add: Suc_times_binomial_eq div_mult_self_is_m zero_less_Suc del: mult_Suc mult_Suc_right) text{*Another version, with -1 instead of Suc.*} lemma times_binomial_minus1_eq: "[|k \ n; 0 (n choose k) * k = n * ((n - 1) choose (k - 1))" apply (cut_tac n = "n - 1" and k = "k - 1" in Suc_times_binomial_eq) apply (simp split add: nat_diff_split, auto) done 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"; val binomial_n_0 = thm"binomial_n_0"; val binomial_0_Suc = thm"binomial_0_Suc"; val binomial_Suc_Suc = thm"binomial_Suc_Suc"; val binomial_eq_0 = thm"binomial_eq_0"; val binomial_n_n = thm"binomial_n_n"; val binomial_Suc_n = thm"binomial_Suc_n"; val binomial_1 = thm"binomial_1"; val zero_less_binomial = thm"zero_less_binomial"; val binomial_eq_0_iff = thm"binomial_eq_0_iff"; val zero_less_binomial_iff = thm"zero_less_binomial_iff"; val Suc_times_binomial_eq = thm"Suc_times_binomial_eq"; val binomial_Suc_Suc_eq_times = thm"binomial_Suc_Suc_eq_times"; val times_binomial_minus1_eq = thm"times_binomial_minus1_eq"; *} end