src/HOL/ex/Sum_of_Powers.thy
 author wenzelm Wed Jun 22 10:09:20 2016 +0200 (2016-06-22) changeset 63343 fb5d8a50c641 parent 61694 6571c78c9667 child 63367 6c731c8b7f03 permissions -rw-r--r--
bundle lifting_syntax;
 lp15@61609 ` 1` ```(* Author: Lukas Bulwahn *) ``` wenzelm@61343 ` 2` ```section \Sum of Powers\ ``` bulwahn@60603 ` 3` bulwahn@60603 ` 4` ```theory Sum_of_Powers ``` bulwahn@60603 ` 5` ```imports Complex_Main ``` bulwahn@60603 ` 6` ```begin ``` bulwahn@60603 ` 7` wenzelm@61343 ` 8` ```subsection \Additions to @{theory Binomial} Theory\ ``` bulwahn@60603 ` 9` bulwahn@60603 ` 10` ```lemma of_nat_binomial_eq_mult_binomial_Suc: ``` bulwahn@60603 ` 11` ``` assumes "k \ n" ``` bulwahn@60603 ` 12` ``` shows "(of_nat :: (nat \ ('a :: field_char_0))) (n choose k) = of_nat (n + 1 - k) / of_nat (n + 1) * of_nat (Suc n choose k)" ``` bulwahn@60603 ` 13` ```proof - ``` bulwahn@60603 ` 14` ``` have "of_nat (n + 1) * (\i 'a)) (n + 1 - k) * (\ii 'a)) (n + 1) * (\i\Suc ` {..i\k. of_nat (Suc n - i))" ``` bulwahn@60603 ` 19` ``` proof (cases k) ``` bulwahn@60603 ` 20` ``` case (Suc k') ``` bulwahn@60603 ` 21` ``` have "of_nat (n + 1) * (\i\Suc ` {..i\insert 0 (Suc ` {..k'}). of_nat (Suc n - i))" ``` bulwahn@60603 ` 22` ``` by (subst setprod.insert) (auto simp add: lessThan_Suc_atMost) ``` bulwahn@60603 ` 23` ``` also have "... = (\i\Suc k'. of_nat (Suc n - i))" by (simp only: Iic_Suc_eq_insert_0) ``` bulwahn@60603 ` 24` ``` finally show ?thesis using Suc by simp ``` bulwahn@60603 ` 25` ``` qed (simp) ``` bulwahn@60603 ` 26` ``` also have "... = (of_nat :: (nat \ 'a)) (Suc n - k) * (\i 'a)) (n + 1 - k) * (\ii 'a)) (n + 1 - k) / of_nat (n + 1) * (\i n" ``` bulwahn@60603 ` 40` ``` shows "(n choose k) = (n + 1 - k) / (n + 1) * (Suc n choose k)" ``` lp15@61609 ` 41` ```by (metis Suc_eq_plus1 add.commute assms le_SucI of_nat_Suc of_nat_binomial_eq_mult_binomial_Suc of_nat_diff) ``` bulwahn@60603 ` 42` wenzelm@61343 ` 43` ```subsection \Preliminaries\ ``` bulwahn@60603 ` 44` bulwahn@60603 ` 45` ```lemma integrals_eq: ``` bulwahn@60603 ` 46` ``` assumes "f 0 = g 0" ``` bulwahn@60603 ` 47` ``` assumes "\ x. ((\x. f x - g x) has_real_derivative 0) (at x)" ``` bulwahn@60603 ` 48` ``` shows "f x = g x" ``` bulwahn@60603 ` 49` ```proof - ``` bulwahn@60603 ` 50` ``` show "f x = g x" ``` bulwahn@60603 ` 51` ``` proof (cases "x \ 0") ``` bulwahn@60603 ` 52` ``` case True ``` bulwahn@60603 ` 53` ``` from assms DERIV_const_ratio_const[OF this, of "\x. f x - g x" 0] ``` bulwahn@60603 ` 54` ``` show ?thesis by auto ``` bulwahn@60603 ` 55` ``` qed (simp add: assms) ``` bulwahn@60603 ` 56` ```qed ``` bulwahn@60603 ` 57` bulwahn@60603 ` 58` ```lemma setsum_diff: "((\i\n::nat. f (i + 1) - f i)::'a::field) = f (n + 1) - f 0" ``` bulwahn@60603 ` 59` ```by (induct n) (auto simp add: field_simps) ``` bulwahn@60603 ` 60` bulwahn@60603 ` 61` ```declare One_nat_def [simp del] ``` bulwahn@60603 ` 62` wenzelm@61343 ` 63` ```subsection \Bernoulli Numbers and Bernoulli Polynomials\ ``` bulwahn@60603 ` 64` bulwahn@60603 ` 65` ```declare setsum.cong [fundef_cong] ``` bulwahn@60603 ` 66` bulwahn@60603 ` 67` ```fun bernoulli :: "nat \ real" ``` bulwahn@60603 ` 68` ```where ``` bulwahn@60603 ` 69` ``` "bernoulli 0 = (1::real)" ``` bulwahn@60603 ` 70` ```| "bernoulli (Suc n) = (-1 / (n + 2)) * (\k \ n. ((n + 2 choose k) * bernoulli k))" ``` bulwahn@60603 ` 71` bulwahn@60603 ` 72` ```declare bernoulli.simps[simp del] ``` bulwahn@60603 ` 73` bulwahn@60603 ` 74` ```definition ``` bulwahn@60603 ` 75` ``` "bernpoly n = (\x. \k \ n. (n choose k) * bernoulli k * x ^ (n - k))" ``` bulwahn@60603 ` 76` wenzelm@61343 ` 77` ```subsection \Basic Observations on Bernoulli Polynomials\ ``` bulwahn@60603 ` 78` bulwahn@60603 ` 79` ```lemma bernpoly_0: "bernpoly n 0 = bernoulli n" ``` bulwahn@60603 ` 80` ```proof (cases n) ``` bulwahn@60603 ` 81` ``` case 0 ``` lp15@61609 ` 82` ``` then show "bernpoly n 0 = bernoulli n" ``` bulwahn@60603 ` 83` ``` unfolding bernpoly_def bernoulli.simps by auto ``` bulwahn@60603 ` 84` ```next ``` bulwahn@60603 ` 85` ``` case (Suc n') ``` bulwahn@60603 ` 86` ``` have "(\k\n'. real (Suc n' choose k) * bernoulli k * 0 ^ (Suc n' - k)) = 0" ``` bulwahn@60603 ` 87` ``` by (rule setsum.neutral) auto ``` bulwahn@60603 ` 88` ``` with Suc show ?thesis ``` bulwahn@60603 ` 89` ``` unfolding bernpoly_def by simp ``` bulwahn@60603 ` 90` ```qed ``` bulwahn@60603 ` 91` bulwahn@60603 ` 92` ```lemma setsum_binomial_times_bernoulli: ``` bulwahn@60603 ` 93` ``` "(\k\n. ((Suc n) choose k) * bernoulli k) = (if n = 0 then 1 else 0)" ``` bulwahn@60603 ` 94` ```proof (cases n) ``` bulwahn@60603 ` 95` ``` case 0 ``` lp15@61609 ` 96` ``` then show ?thesis by (simp add: bernoulli.simps) ``` bulwahn@60603 ` 97` ```next ``` bulwahn@60603 ` 98` ``` case Suc ``` lp15@61609 ` 99` ``` then show ?thesis ``` bulwahn@60603 ` 100` ``` by (simp add: bernoulli.simps) ``` bulwahn@60603 ` 101` ``` (simp add: field_simps add_2_eq_Suc'[symmetric] del: add_2_eq_Suc add_2_eq_Suc') ``` bulwahn@60603 ` 102` ```qed ``` bulwahn@60603 ` 103` wenzelm@61343 ` 104` ```subsection \Sum of Powers with Bernoulli Polynomials\ ``` bulwahn@60603 ` 105` bulwahn@60603 ` 106` ```lemma bernpoly_derivative [derivative_intros]: ``` bulwahn@60603 ` 107` ``` "(bernpoly (Suc n) has_real_derivative ((n + 1) * bernpoly n x)) (at x)" ``` bulwahn@60603 ` 108` ```proof - ``` bulwahn@60603 ` 109` ``` have "(bernpoly (Suc n) has_real_derivative (\k\n. real (Suc n - k) * x ^ (n - k) * (real (Suc n choose k) * bernoulli k))) (at x)" ``` bulwahn@60603 ` 110` ``` unfolding bernpoly_def by (rule DERIV_cong) (fast intro!: derivative_intros, simp) ``` bulwahn@60603 ` 111` ``` moreover have "(\k\n. real (Suc n - k) * x ^ (n - k) * (real (Suc n choose k) * bernoulli k)) = (n + 1) * bernpoly n x" ``` bulwahn@60603 ` 112` ``` unfolding bernpoly_def ``` lp15@61609 ` 113` ``` by (auto intro: setsum.cong simp add: setsum_right_distrib real_binomial_eq_mult_binomial_Suc[of _ n] Suc_eq_plus1 of_nat_diff) ``` bulwahn@60603 ` 114` ``` ultimately show ?thesis by auto ``` bulwahn@60603 ` 115` ```qed ``` bulwahn@60603 ` 116` bulwahn@60603 ` 117` ```lemma diff_bernpoly: ``` bulwahn@60603 ` 118` ``` "bernpoly n (x + 1) - bernpoly n x = n * x ^ (n - 1)" ``` bulwahn@60603 ` 119` ```proof (induct n arbitrary: x) ``` bulwahn@60603 ` 120` ``` case 0 ``` bulwahn@60603 ` 121` ``` show ?case unfolding bernpoly_def by auto ``` bulwahn@60603 ` 122` ```next ``` bulwahn@60603 ` 123` ``` case (Suc n) ``` bulwahn@60603 ` 124` ``` have "bernpoly (Suc n) (0 + 1) - bernpoly (Suc n) 0 = (Suc n) * 0 ^ n" ``` bulwahn@60603 ` 125` ``` unfolding bernpoly_0 unfolding bernpoly_def by (simp add: setsum_binomial_times_bernoulli zero_power) ``` lp15@61609 ` 126` ``` then have const: "bernpoly (Suc n) (0 + 1) - bernpoly (Suc n) 0 = real (Suc n) * 0 ^ n" by (simp add: power_0_left) ``` bulwahn@60603 ` 127` ``` have hyps': "\x. (real n + 1) * bernpoly n (x + 1) - (real n + 1) * bernpoly n x = real n * x ^ (n - Suc 0) * real (Suc n)" ``` bulwahn@60603 ` 128` ``` unfolding right_diff_distrib[symmetric] by (simp add: Suc.hyps One_nat_def) ``` bulwahn@60603 ` 129` ``` note [derivative_intros] = DERIV_chain'[where f = "\x::real. x + 1" and g = "bernpoly (Suc n)" and s="UNIV"] ``` bulwahn@60603 ` 130` ``` have derivative: "\x. ((%x. bernpoly (Suc n) (x + 1) - bernpoly (Suc n) x - real (Suc n) * x ^ n) has_real_derivative 0) (at x)" ``` bulwahn@60603 ` 131` ``` by (rule DERIV_cong) (fast intro!: derivative_intros, simp add: hyps') ``` bulwahn@60603 ` 132` ``` from integrals_eq[OF const derivative] show ?case by simp ``` bulwahn@60603 ` 133` ```qed ``` bulwahn@60603 ` 134` bulwahn@60603 ` 135` ```lemma sum_of_powers: "(\k\n::nat. (real k) ^ m) = (bernpoly (Suc m) (n + 1) - bernpoly (Suc m) 0) / (m + 1)" ``` bulwahn@60603 ` 136` ```proof - ``` bulwahn@60603 ` 137` ``` from diff_bernpoly[of "Suc m", simplified] have "(m + (1::real)) * (\k\n. (real k) ^ m) = (\k\n. bernpoly (Suc m) (real k + 1) - bernpoly (Suc m) (real k))" ``` bulwahn@60603 ` 138` ``` by (auto simp add: setsum_right_distrib intro!: setsum.cong) ``` bulwahn@60603 ` 139` ``` also have "... = (\k\n. bernpoly (Suc m) (real (k + 1)) - bernpoly (Suc m) (real k))" ``` lp15@61609 ` 140` ``` by simp ``` bulwahn@60603 ` 141` ``` also have "... = bernpoly (Suc m) (n + 1) - bernpoly (Suc m) 0" ``` bulwahn@60603 ` 142` ``` by (simp only: setsum_diff[where f="\k. bernpoly (Suc m) (real k)"]) simp ``` bulwahn@60603 ` 143` ``` finally show ?thesis by (auto simp add: field_simps intro!: eq_divide_imp) ``` bulwahn@60603 ` 144` ```qed ``` bulwahn@60603 ` 145` wenzelm@61343 ` 146` ```subsection \Instances for Square And Cubic Numbers\ ``` bulwahn@60603 ` 147` bulwahn@60603 ` 148` ```lemma binomial_unroll: ``` bulwahn@60603 ` 149` ``` "n > 0 \ (n choose k) = (if k = 0 then 1 else (n - 1) choose (k - 1) + ((n - 1) choose k))" ``` bulwahn@60603 ` 150` ```by (cases n) (auto simp add: binomial.simps(2)) ``` bulwahn@60603 ` 151` bulwahn@60603 ` 152` ```lemma setsum_unroll: ``` bulwahn@60603 ` 153` ``` "(\k\n::nat. f k) = (if n = 0 then f 0 else f n + (\k\n - 1. f k))" ``` bulwahn@60603 ` 154` ```by auto (metis One_nat_def Suc_pred add.commute setsum_atMost_Suc) ``` bulwahn@60603 ` 155` bulwahn@60603 ` 156` ```lemma bernoulli_unroll: ``` bulwahn@60603 ` 157` ``` "n > 0 \ bernoulli n = - 1 / (real n + 1) * (\k\n - 1. real (n + 1 choose k) * bernoulli k)" ``` bulwahn@60603 ` 158` ```by (cases n) (simp add: bernoulli.simps One_nat_def)+ ``` bulwahn@60603 ` 159` bulwahn@60603 ` 160` ```lemmas unroll = binomial.simps(1) binomial_unroll ``` bulwahn@60603 ` 161` ``` bernoulli.simps(1) bernoulli_unroll setsum_unroll bernpoly_def ``` bulwahn@60603 ` 162` bulwahn@60603 ` 163` ```lemma sum_of_squares: "(\k\n::nat. k ^ 2) = (2 * n ^ 3 + 3 * n ^ 2 + n) / 6" ``` bulwahn@60603 ` 164` ```proof - ``` bulwahn@60603 ` 165` ``` have "real (\k\n::nat. k ^ 2) = (\k\n::nat. (real k) ^ 2)" by simp ``` bulwahn@60603 ` 166` ``` also have "... = (bernpoly 3 (real (n + 1)) - bernpoly 3 0) / real (3 :: nat)" ``` bulwahn@60603 ` 167` ``` by (auto simp add: sum_of_powers) ``` bulwahn@60603 ` 168` ``` also have "... = (2 * n ^ 3 + 3 * n ^ 2 + n) / 6" ``` bulwahn@60603 ` 169` ``` by (simp add: unroll algebra_simps power2_eq_square power3_eq_cube One_nat_def[symmetric]) ``` bulwahn@60603 ` 170` ``` finally show ?thesis by simp ``` bulwahn@60603 ` 171` ```qed ``` bulwahn@60603 ` 172` bulwahn@60603 ` 173` ```lemma sum_of_squares_nat: "(\k\n::nat. k ^ 2) = (2 * n ^ 3 + 3 * n ^ 2 + n) div 6" ``` bulwahn@60603 ` 174` ```proof - ``` bulwahn@60603 ` 175` ``` from sum_of_squares have "real (6 * (\k\n. k ^ 2)) = real (2 * n ^ 3 + 3 * n ^ 2 + n)" ``` bulwahn@60603 ` 176` ``` by (auto simp add: field_simps) ``` lp15@61609 ` 177` ``` then have "6 * (\k\n. k ^ 2) = 2 * n ^ 3 + 3 * n ^ 2 + n" ``` lp15@61649 ` 178` ``` using of_nat_eq_iff by blast ``` lp15@61609 ` 179` ``` then show ?thesis by auto ``` bulwahn@60603 ` 180` ```qed ``` bulwahn@60603 ` 181` bulwahn@60603 ` 182` ```lemma sum_of_cubes: "(\k\n::nat. k ^ 3) = (n ^ 2 + n) ^ 2 / 4" ``` bulwahn@60603 ` 183` ```proof - ``` bulwahn@60603 ` 184` ``` have two_plus_two: "2 + 2 = 4" by simp ``` bulwahn@60603 ` 185` ``` have power4_eq: "\x::real. x ^ 4 = x * x * x * x" ``` bulwahn@60603 ` 186` ``` by (simp only: two_plus_two[symmetric] power_add power2_eq_square) ``` bulwahn@60603 ` 187` ``` have "real (\k\n::nat. k ^ 3) = (\k\n::nat. (real k) ^ 3)" by simp ``` bulwahn@60603 ` 188` ``` also have "... = ((bernpoly 4 (n + 1) - bernpoly 4 0)) / (real (4 :: nat))" ``` bulwahn@60603 ` 189` ``` by (auto simp add: sum_of_powers) ``` bulwahn@60603 ` 190` ``` also have "... = ((n ^ 2 + n) / 2) ^ 2" ``` bulwahn@60603 ` 191` ``` by (simp add: unroll algebra_simps power2_eq_square power4_eq power3_eq_cube) ``` lp15@61694 ` 192` ``` finally show ?thesis by (simp add: power_divide) ``` bulwahn@60603 ` 193` ```qed ``` lp15@61609 ` 194` ``` ``` bulwahn@60603 ` 195` ```lemma sum_of_cubes_nat: "(\k\n::nat. k ^ 3) = (n ^ 2 + n) ^ 2 div 4" ``` bulwahn@60603 ` 196` ```proof - ``` bulwahn@60603 ` 197` ``` from sum_of_cubes have "real (4 * (\k\n. k ^ 3)) = real ((n ^ 2 + n) ^ 2)" ``` bulwahn@60603 ` 198` ``` by (auto simp add: field_simps) ``` lp15@61609 ` 199` ``` then have "4 * (\k\n. k ^ 3) = (n ^ 2 + n) ^ 2" ``` lp15@61649 ` 200` ``` using of_nat_eq_iff by blast ``` lp15@61609 ` 201` ``` then show ?thesis by auto ``` bulwahn@60603 ` 202` ```qed ``` bulwahn@60603 ` 203` bulwahn@60603 ` 204` ```end ```