(* Author: Lukas Bulwahn <lukas.bulwahn-at-gmail.com> *)
section \<open>Sum of Powers\<close>
theory Sum_of_Powers
imports Complex_Main
begin
subsection \<open>Additions to @{theory Binomial} Theory\<close>
lemma of_nat_binomial_eq_mult_binomial_Suc:
assumes "k \<le> n"
shows "(of_nat :: (nat \<Rightarrow> ('a :: field_char_0))) (n choose k) = of_nat (n + 1 - k) / of_nat (n + 1) * of_nat (Suc n choose k)"
proof -
have "of_nat (n + 1) * (\<Prod>i<k. of_nat (n - i)) = (of_nat :: (nat \<Rightarrow> 'a)) (n + 1 - k) * (\<Prod>i<k. of_nat (Suc n - i))"
proof -
have "of_nat (n + 1) * (\<Prod>i<k. of_nat (n - i)) = (of_nat :: (nat \<Rightarrow> 'a)) (n + 1) * (\<Prod>i\<in>Suc ` {..<k}. of_nat (Suc n - i))"
by (auto simp add: setprod.reindex)
also have "... = (\<Prod>i\<le>k. of_nat (Suc n - i))"
proof (cases k)
case (Suc k')
have "of_nat (n + 1) * (\<Prod>i\<in>Suc ` {..<Suc k'}. of_nat (Suc n - i)) = (\<Prod>i\<in>insert 0 (Suc ` {..k'}). of_nat (Suc n - i))"
by (subst setprod.insert) (auto simp add: lessThan_Suc_atMost)
also have "... = (\<Prod>i\<le>Suc k'. of_nat (Suc n - i))" by (simp only: Iic_Suc_eq_insert_0)
finally show ?thesis using Suc by simp
qed (simp)
also have "... = (of_nat :: (nat \<Rightarrow> 'a)) (Suc n - k) * (\<Prod>i<k. of_nat (Suc n - i))"
by (cases k) (auto simp add: atMost_Suc lessThan_Suc_atMost)
also have "... = (of_nat :: (nat \<Rightarrow> 'a)) (n + 1 - k) * (\<Prod>i<k. of_nat (Suc n - i))"
by (simp only: Suc_eq_plus1)
finally show ?thesis .
qed
then have "(\<Prod>i<k. of_nat (n - i)) = (of_nat :: (nat \<Rightarrow> 'a)) (n + 1 - k) / of_nat (n + 1) * (\<Prod>i<k. of_nat (Suc n - i))"
by (metis (no_types, lifting) le_add2 nonzero_mult_divide_cancel_left not_one_le_zero of_nat_eq_0_iff times_divide_eq_left)
from assms this show ?thesis
by (auto simp add: binomial_altdef_of_nat setprod_dividef)
qed
lemma real_binomial_eq_mult_binomial_Suc:
assumes "k \<le> n"
shows "(n choose k) = (n + 1 - k) / (n + 1) * (Suc n choose k)"
by (metis Suc_eq_plus1 add.commute assms le_SucI of_nat_Suc of_nat_binomial_eq_mult_binomial_Suc of_nat_diff)
subsection \<open>Preliminaries\<close>
lemma integrals_eq:
assumes "f 0 = g 0"
assumes "\<And> x. ((\<lambda>x. f x - g x) has_real_derivative 0) (at x)"
shows "f x = g x"
proof -
show "f x = g x"
proof (cases "x \<noteq> 0")
case True
from assms DERIV_const_ratio_const[OF this, of "\<lambda>x. f x - g x" 0]
show ?thesis by auto
qed (simp add: assms)
qed
lemma setsum_diff: "((\<Sum>i\<le>n::nat. f (i + 1) - f i)::'a::field) = f (n + 1) - f 0"
by (induct n) (auto simp add: field_simps)
declare One_nat_def [simp del]
subsection \<open>Bernoulli Numbers and Bernoulli Polynomials\<close>
declare setsum.cong [fundef_cong]
fun bernoulli :: "nat \<Rightarrow> real"
where
"bernoulli 0 = (1::real)"
| "bernoulli (Suc n) = (-1 / (n + 2)) * (\<Sum>k \<le> n. ((n + 2 choose k) * bernoulli k))"
declare bernoulli.simps[simp del]
definition
"bernpoly n = (\<lambda>x. \<Sum>k \<le> n. (n choose k) * bernoulli k * x ^ (n - k))"
subsection \<open>Basic Observations on Bernoulli Polynomials\<close>
lemma bernpoly_0: "bernpoly n 0 = bernoulli n"
proof (cases n)
case 0
then show "bernpoly n 0 = bernoulli n"
unfolding bernpoly_def bernoulli.simps by auto
next
case (Suc n')
have "(\<Sum>k\<le>n'. real (Suc n' choose k) * bernoulli k * 0 ^ (Suc n' - k)) = 0"
by (rule setsum.neutral) auto
with Suc show ?thesis
unfolding bernpoly_def by simp
qed
lemma setsum_binomial_times_bernoulli:
"(\<Sum>k\<le>n. ((Suc n) choose k) * bernoulli k) = (if n = 0 then 1 else 0)"
proof (cases n)
case 0
then show ?thesis by (simp add: bernoulli.simps)
next
case Suc
then show ?thesis
by (simp add: bernoulli.simps)
(simp add: field_simps add_2_eq_Suc'[symmetric] del: add_2_eq_Suc add_2_eq_Suc')
qed
subsection \<open>Sum of Powers with Bernoulli Polynomials\<close>
lemma bernpoly_derivative [derivative_intros]:
"(bernpoly (Suc n) has_real_derivative ((n + 1) * bernpoly n x)) (at x)"
proof -
have "(bernpoly (Suc n) has_real_derivative (\<Sum>k\<le>n. real (Suc n - k) * x ^ (n - k) * (real (Suc n choose k) * bernoulli k))) (at x)"
unfolding bernpoly_def by (rule DERIV_cong) (fast intro!: derivative_intros, simp)
moreover have "(\<Sum>k\<le>n. real (Suc n - k) * x ^ (n - k) * (real (Suc n choose k) * bernoulli k)) = (n + 1) * bernpoly n x"
unfolding bernpoly_def
by (auto intro: setsum.cong simp add: setsum_right_distrib real_binomial_eq_mult_binomial_Suc[of _ n] Suc_eq_plus1 of_nat_diff)
ultimately show ?thesis by auto
qed
lemma diff_bernpoly:
"bernpoly n (x + 1) - bernpoly n x = n * x ^ (n - 1)"
proof (induct n arbitrary: x)
case 0
show ?case unfolding bernpoly_def by auto
next
case (Suc n)
have "bernpoly (Suc n) (0 + 1) - bernpoly (Suc n) 0 = (Suc n) * 0 ^ n"
unfolding bernpoly_0 unfolding bernpoly_def by (simp add: setsum_binomial_times_bernoulli zero_power)
then have const: "bernpoly (Suc n) (0 + 1) - bernpoly (Suc n) 0 = real (Suc n) * 0 ^ n" by (simp add: power_0_left)
have hyps': "\<And>x. (real n + 1) * bernpoly n (x + 1) - (real n + 1) * bernpoly n x = real n * x ^ (n - Suc 0) * real (Suc n)"
unfolding right_diff_distrib[symmetric] by (simp add: Suc.hyps One_nat_def)
note [derivative_intros] = DERIV_chain'[where f = "\<lambda>x::real. x + 1" and g = "bernpoly (Suc n)" and s="UNIV"]
have derivative: "\<And>x. ((%x. bernpoly (Suc n) (x + 1) - bernpoly (Suc n) x - real (Suc n) * x ^ n) has_real_derivative 0) (at x)"
by (rule DERIV_cong) (fast intro!: derivative_intros, simp add: hyps')
from integrals_eq[OF const derivative] show ?case by simp
qed
lemma sum_of_powers: "(\<Sum>k\<le>n::nat. (real k) ^ m) = (bernpoly (Suc m) (n + 1) - bernpoly (Suc m) 0) / (m + 1)"
proof -
from diff_bernpoly[of "Suc m", simplified] have "(m + (1::real)) * (\<Sum>k\<le>n. (real k) ^ m) = (\<Sum>k\<le>n. bernpoly (Suc m) (real k + 1) - bernpoly (Suc m) (real k))"
by (auto simp add: setsum_right_distrib intro!: setsum.cong)
also have "... = (\<Sum>k\<le>n. bernpoly (Suc m) (real (k + 1)) - bernpoly (Suc m) (real k))"
by simp
also have "... = bernpoly (Suc m) (n + 1) - bernpoly (Suc m) 0"
by (simp only: setsum_diff[where f="\<lambda>k. bernpoly (Suc m) (real k)"]) simp
finally show ?thesis by (auto simp add: field_simps intro!: eq_divide_imp)
qed
subsection \<open>Instances for Square And Cubic Numbers\<close>
lemma binomial_unroll:
"n > 0 \<Longrightarrow> (n choose k) = (if k = 0 then 1 else (n - 1) choose (k - 1) + ((n - 1) choose k))"
by (auto simp add: gr0_conv_Suc)
lemma setsum_unroll:
"(\<Sum>k\<le>n::nat. f k) = (if n = 0 then f 0 else f n + (\<Sum>k\<le>n - 1. f k))"
by auto (metis One_nat_def Suc_pred add.commute setsum_atMost_Suc)
lemma bernoulli_unroll:
"n > 0 \<Longrightarrow> bernoulli n = - 1 / (real n + 1) * (\<Sum>k\<le>n - 1. real (n + 1 choose k) * bernoulli k)"
by (cases n) (simp add: bernoulli.simps One_nat_def)+
lemmas unroll = binomial_unroll
bernoulli.simps(1) bernoulli_unroll setsum_unroll bernpoly_def
lemma sum_of_squares: "(\<Sum>k\<le>n::nat. k ^ 2) = (2 * n ^ 3 + 3 * n ^ 2 + n) / 6"
proof -
have "real (\<Sum>k\<le>n::nat. k ^ 2) = (\<Sum>k\<le>n::nat. (real k) ^ 2)" by simp
also have "... = (bernpoly 3 (real (n + 1)) - bernpoly 3 0) / real (3 :: nat)"
by (auto simp add: sum_of_powers)
also have "... = (2 * n ^ 3 + 3 * n ^ 2 + n) / 6"
by (simp add: unroll algebra_simps power2_eq_square power3_eq_cube One_nat_def[symmetric])
finally show ?thesis by simp
qed
lemma sum_of_squares_nat: "(\<Sum>k\<le>n::nat. k ^ 2) = (2 * n ^ 3 + 3 * n ^ 2 + n) div 6"
proof -
from sum_of_squares have "real (6 * (\<Sum>k\<le>n. k ^ 2)) = real (2 * n ^ 3 + 3 * n ^ 2 + n)"
by (auto simp add: field_simps)
then have "6 * (\<Sum>k\<le>n. k ^ 2) = 2 * n ^ 3 + 3 * n ^ 2 + n"
using of_nat_eq_iff by blast
then show ?thesis by auto
qed
lemma sum_of_cubes: "(\<Sum>k\<le>n::nat. k ^ 3) = (n ^ 2 + n) ^ 2 / 4"
proof -
have two_plus_two: "2 + 2 = 4" by simp
have power4_eq: "\<And>x::real. x ^ 4 = x * x * x * x"
by (simp only: two_plus_two[symmetric] power_add power2_eq_square)
have "real (\<Sum>k\<le>n::nat. k ^ 3) = (\<Sum>k\<le>n::nat. (real k) ^ 3)" by simp
also have "... = ((bernpoly 4 (n + 1) - bernpoly 4 0)) / (real (4 :: nat))"
by (auto simp add: sum_of_powers)
also have "... = ((n ^ 2 + n) / 2) ^ 2"
by (simp add: unroll algebra_simps power2_eq_square power4_eq power3_eq_cube)
finally show ?thesis by (simp add: power_divide)
qed
lemma sum_of_cubes_nat: "(\<Sum>k\<le>n::nat. k ^ 3) = (n ^ 2 + n) ^ 2 div 4"
proof -
from sum_of_cubes have "real (4 * (\<Sum>k\<le>n. k ^ 3)) = real ((n ^ 2 + n) ^ 2)"
by (auto simp add: field_simps)
then have "4 * (\<Sum>k\<le>n. k ^ 3) = (n ^ 2 + n) ^ 2"
using of_nat_eq_iff by blast
then show ?thesis by auto
qed
end