# Theory NatSum

theory NatSum
imports Main
```(*  Title:  HOL/ex/NatSum.thy
Author: Tobias Nipkow
*)

section ‹Summing natural numbers›

theory NatSum imports Main begin

text ‹
Summing natural numbers, squares, cubes, etc.

Thanks to Sloane's On-Line Encyclopedia of Integer Sequences,
🌐‹http://www.research.att.com/~njas/sequences›.
›

lemmas [simp] =
ring_distribs
diff_mult_distrib diff_mult_distrib2 ―‹for type nat›

text ‹
\medskip The sum of the first ‹n› odd numbers equals ‹n›
squared.
›

lemma sum_of_odds: "(∑i=0..<n. Suc (i + i)) = n * n"
by (induct n) auto

text ‹
\medskip The sum of the first ‹n› odd squares.
›

lemma sum_of_odd_squares:
"3 * (∑i=0..<n. Suc(2*i) * Suc(2*i)) = n * (4 * n * n - 1)"
by (induct n) auto

text ‹
\medskip The sum of the first ‹n› odd cubes
›

lemma sum_of_odd_cubes:
"(∑i=0..<n. Suc (2*i) * Suc (2*i) * Suc (2*i)) =
n * n * (2 * n * n - 1)"
by (induct n) auto

text ‹
\medskip The sum of the first ‹n› positive integers equals
‹n (n + 1) / 2›.›

lemma sum_of_naturals:
"2 * (∑i=0..n. i) = n * Suc n"
by (induct n) auto

lemma sum_of_squares:
"6 * (∑i=0..n. i * i) = n * Suc n * Suc (2 * n)"
by (induct n) auto

lemma sum_of_cubes:
"4 * (∑i=0..n. i * i * i) = n * n * Suc n * Suc n"
by (induct n) auto

text‹\medskip A cute identity:›

lemma sum_squared: "(∑i=0..n. i)^2 = (∑i=0..n::nat. i^3)"
proof(induct n)
case 0 show ?case by simp
next
case (Suc n)
have "(∑i = 0..Suc n. i)^2 =
(∑i = 0..n. i^3) + (2*(∑i = 0..n. i)*(n+1) + (n+1)^2)"
(is "_ = ?A + ?B")
also have "?B = (n+1)^3"
also have "?A + (n+1)^3 = (∑i=0..Suc n. i^3)" by simp
finally show ?case .
qed

text ‹
\medskip Sum of fourth powers: three versions.
›

lemma sum_of_fourth_powers:
"30 * (∑i=0..n. i * i * i * i) =
n * Suc n * Suc (2 * n) * (3 * n * n + 3 * n - 1)"
apply (induct n)
apply simp_all
apply (case_tac n)  ― ‹eliminates the subtraction›
apply (simp_all (no_asm_simp))
done

text ‹
Two alternative proofs, with a change of variables and much more
subtraction, performed using the integers.›

lemma int_sum_of_fourth_powers:
"30 * int (∑i=0..<m. i * i * i * i) =
int m * (int m - 1) * (int(2 * m) - 1) *
(int(3 * m * m) - int(3 * m) - 1)"
by (induct m) (simp_all add: of_nat_mult)

lemma of_nat_sum_of_fourth_powers:
"30 * of_nat (∑i=0..<m. i * i * i * i) =
of_nat m * (of_nat m - 1) * (of_nat (2 * m) - 1) *
(of_nat (3 * m * m) - of_nat (3 * m) - (1::int))"
by (induct m) (simp_all add: of_nat_mult)

text ‹
\medskip Sums of geometric series: ‹2›, ‹3› and the
general case.
›

lemma sum_of_2_powers: "(∑i=0..<n. 2^i) = 2^n - (1::nat)"
by (induct n) (auto split: nat_diff_split)

lemma sum_of_3_powers: "2 * (∑i=0..<n. 3^i) = 3^n - (1::nat)"
by (induct n) auto

lemma sum_of_powers: "0 < k ==> (k - 1) * (∑i=0..<n. k^i) = k^n - (1::nat)"
by (induct n) auto

end
```