src/HOL/ex/NatSum.thy
 author wenzelm Thu, 01 Feb 2001 20:51:13 +0100 changeset 11024 23bf8d787b04 parent 8944 96964d43a472 child 11377 0f16ad464c62 permissions -rw-r--r--
converted to new-style theories;
```
(*  Title:      HOL/ex/NatSum.ML
ID:         \$Id\$
Author:     Tobias Nipkow

Summing natural numbers, squares, cubes, etc.

Originally demonstrated permutative rewriting, but add_ac is no longer
needed thanks to new simprocs.

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

header {* Summing natural numbers *}

theory NatSum = Main:

declare lessThan_Suc [simp] atMost_Suc [simp]
declare diff_mult_distrib [simp] diff_mult_distrib2 [simp]

text {*
\medskip The sum of the first @{term n} odd numbers equals @{term n}
squared.
*}

lemma sum_of_odds: "setsum (\<lambda>i. Suc (i + i)) (lessThan n) = 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 * setsum (\<lambda>i. Suc (i + i) * Suc (i + i)) (lessThan n) = n * (#4 * n * n - #1)"
apply (induct n)
txt {* This removes the @{term "-#1"} from the inductive step *}
apply (case_tac [2] n)
apply auto
done

text {*
\medskip The sum of the first @{term n} odd cubes
*}

lemma sum_of_odd_cubes:
"setsum (\<lambda>i. Suc (i + i) * Suc (i + i) * Suc (i + i)) (lessThan n) =
n * n * (#2 * n * n - #1)"
apply (induct "n")
txt {* This removes the @{term "-#1"} from the inductive step *}
apply (case_tac [2] "n")
apply auto
done

text {*
\medskip The sum of the first @{term n} positive integers equals
@{text "n (n + 1) / 2"}.*}

lemma sum_of_naturals: "#2 * setsum id (atMost n) = n * Suc n"
apply (induct n)
apply auto
done

lemma sum_of_squares: "#6 * setsum (\<lambda>i. i * i) (atMost n) = n * Suc n * Suc (#2 * n)"
apply (induct n)
apply auto
done

lemma sum_of_cubes: "#4 * setsum (\<lambda>i. i * i * i) (atMost n) = 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 * setsum (\<lambda>i. i * i * i * i) (atMost n) =
n * Suc n * Suc (#2 * n) * (#3 * n * n + #3 * n - #1)"
apply (induct n)
apply auto
txt {* Eliminates the subtraction *}
apply (case_tac n)
apply simp_all
done

text {*
Alternative proof, with a change of variables and much more
subtraction, performed using the integers. *}

declare
zmult_int [symmetric, simp]
zdiff_zmult_distrib [simp]
zdiff_zmult_distrib2 [simp]

lemma int_sum_of_fourth_powers:
"#30 * int (setsum (\<lambda>i. i * i * i * i) (lessThan m)) =
int m * (int m - #1) * (int (#2 * m) - #1) *
(int (#3 * m * m) - int (#3 * m) - #1)"
apply (induct m)
apply simp_all
done

text {*
\medskip Sums of geometric series: 2, 3 and the general case *}

lemma sum_of_2_powers: "setsum (\<lambda>i. #2^i) (lessThan n) = #2^n - 1"
apply (induct n)
apply auto
done

lemma sum_of_3_powers: "#2 * setsum (\<lambda>i. #3^i) (lessThan n) = #3^n - 1"
apply (induct n)
apply auto
done

lemma sum_of_powers: "0 < k ==> (k - 1) * setsum (\<lambda>i. k^i) (lessThan n) = k^n - 1"
apply (induct n)
apply auto
done

end
```