--- a/src/HOL/ex/NatSum.thy Tue Jun 28 16:12:03 2005 +0200
+++ b/src/HOL/ex/NatSum.thy Tue Jun 28 17:56:04 2005 +0200
@@ -18,7 +18,6 @@
*}
lemmas [simp] =
- lessThan_Suc atMost_Suc setsum_op_ivl_Suc setsum_cl_ivl_Suc
left_distrib right_distrib
left_diff_distrib right_diff_distrib --{*for true subtraction*}
diff_mult_distrib diff_mult_distrib2 --{*for type nat*}
@@ -28,10 +27,8 @@
squared.
*}
-lemma sum_of_odds: "(\<Sum>i \<in> {0..<n}. Suc (i + i)) = n * n"
- apply (induct n)
- apply auto
- done
+lemma sum_of_odds: "(\<Sum>i=0..<n. Suc (i + i)) = n * n"
+ by (induct n, auto)
text {*
@@ -40,23 +37,17 @@
lemma sum_of_odd_squares:
"3 * (\<Sum>i=0..<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
+ by (induct n, auto)
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=0..<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
+ by (induct n, auto)
text {*
\medskip The sum of the first @{text n} positive integers equals
@@ -64,21 +55,15 @@
lemma sum_of_naturals:
"2 * (\<Sum>i=0..n. i) = n * Suc n"
- apply (induct n)
- apply auto
- done
+ by (induct n, auto)
lemma sum_of_squares:
"6 * (\<Sum>i=0..n. i * i) = n * Suc n * Suc (2 * n)"
- apply (induct n)
- apply auto
- done
+ by (induct n, auto)
lemma sum_of_cubes:
"4 * (\<Sum>i=0..n. i * i * i) = n * n * Suc n * Suc n"
- apply (induct n)
- apply auto
- done
+ by (induct n, auto)
text {*
@@ -95,24 +80,20 @@
done
text {*
- Tow alternative proofs, with a change of variables and much more
+ Two alternative proofs, with a change of variables and much more
subtraction, performed using the integers. *}
lemma int_sum_of_fourth_powers:
"30 * int (\<Sum>i=0..<m. i * i * i * i) =
int m * (int m - 1) * (int(2 * m) - 1) *
(int(3 * m * m) - int(3 * m) - 1)"
- apply (induct m)
- apply (simp_all add:zmult_int[symmetric])
- done
+ by (induct m, simp_all add: int_mult)
lemma of_nat_sum_of_fourth_powers:
"30 * of_nat (\<Sum>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))"
- apply (induct m)
- apply simp_all
- done
+ by (induct m, simp_all)
text {*
@@ -126,13 +107,10 @@
done
lemma sum_of_3_powers: "2 * (\<Sum>i=0..<n. 3^i) = 3^n - (1::nat)"
- apply (induct n)
- apply auto
- done
+ by (induct n, auto)
lemma sum_of_powers: "0 < k ==> (k - 1) * (\<Sum>i=0..<n. k^i) = k^n - (1::nat)"
- apply (induct n)
- apply auto
- done
+ by (induct n, auto)
+
end