(* Title: HOL/ex/NatSum.ML
ID: $Id$
Author: Tobias Nipkow
*)
header {* Summing natural numbers *}
theory NatSum = Main:
text {*
Summing natural numbers, squares, cubes, etc.
Originally demonstrated permutative rewriting, but @{thm [source]
add_ac} is no longer needed thanks to new simprocs.
Thanks to Sloane's On-Line Encyclopedia of Integer Sequences,
\url{http://www.research.att.com/~njas/sequences/}.
*}
lemmas [simp] = lessThan_Suc atMost_Suc left_distrib right_distrib
left_diff_distrib right_diff_distrib --{*for true subtraction*}
diff_mult_distrib diff_mult_distrib2 --{*for type nat*}
text {*
\medskip The sum of the first @{text n} odd numbers equals @{text n}
squared.
*}
lemma sum_of_odds: "(\<Sum>i \<in> {..<n}. Suc (i + i)) = n * n"
apply (induct n)
apply auto
done
text {*
\medskip The sum of the first @{text n} odd squares.
*}
lemma sum_of_odd_squares:
"3 * (\<Sum>i \<in> {..<n}. Suc (2*i) * Suc (2*i)) =
n * (4 * n * n - 1)"
apply (induct n)
apply (auto split: nat_diff_split) (*eliminate the subtraction*)
done
text {*
\medskip The sum of the first @{text n} odd cubes
*}
lemma numeral_2_eq_2: "2 = Suc (Suc 0)" by (auto );
lemma sum_of_odd_cubes:
"(\<Sum>i \<in> {..<n}. Suc (2*i) * Suc (2*i) * Suc (2*i)) =
n * n * (2 * n * n - 1)"
apply (induct n)
apply (auto split: nat_diff_split) (*eliminate the subtraction*)
done
text {*
\medskip The sum of the first @{text n} positive integers equals
@{text "n (n + 1) / 2"}.*}
lemma sum_of_naturals:
"2 * (\<Sum>i \<in> {..n}. i) = n * Suc n"
apply (induct n)
apply auto
done
lemma sum_of_squares:
"6 * (\<Sum>i \<in> {..n}. i * i) = n * Suc n * Suc (2 * n)"
apply (induct n)
apply auto
done
lemma sum_of_cubes:
"4 * (\<Sum>i \<in> {..n}. i * i * i) = n * n * Suc n * Suc n"
apply (induct n)
apply auto
done
text {*
\medskip Sum of fourth powers: two versions.
*}
lemma sum_of_fourth_powers:
"30 * (\<Sum>i \<in> {..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 {*
Alternative proof, with a change of variables and much more
subtraction, performed using the integers. *}
lemma int_sum_of_fourth_powers:
"30 * of_nat (\<Sum>i \<in> {..<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))"
apply (induct m)
apply simp_all
done
text {*
\medskip Sums of geometric series: @{text 2}, @{text 3} and the
general case.
*}
lemma sum_of_2_powers: "(\<Sum>i \<in> {..<n}. 2^i) = 2^n - (1::nat)"
apply (induct n)
apply (auto split: nat_diff_split)
done
lemma sum_of_3_powers: "2 * (\<Sum>i \<in> {..<n}. 3^i) = 3^n - (1::nat)"
apply (induct n)
apply auto
done
lemma sum_of_powers: "0 < k ==> (k - 1) * (\<Sum>i \<in> {..<n}. k^i) = k^n - (1::nat)"
apply (induct n)
apply auto
done
end