--- a/src/HOL/ex/NatSum.ML Thu Nov 20 11:54:31 1997 +0100
+++ b/src/HOL/ex/NatSum.ML Thu Nov 20 11:55:39 1997 +0100
@@ -13,7 +13,7 @@
(*The sum of the first n positive integers equals n(n+1)/2.*)
goal NatSum.thy "2*sum (%i. i) (Suc n) = n*Suc(n)";
by (Simp_tac 1);
-by (nat_ind_tac "n" 1);
+by (induct_tac "n" 1);
by (Simp_tac 1);
by (Asm_simp_tac 1);
qed "sum_of_naturals";
@@ -21,7 +21,7 @@
goal NatSum.thy
"Suc(Suc(Suc(Suc 2)))*sum (%i. i*i) (Suc n) = n*Suc(n)*Suc(2*n)";
by (Simp_tac 1);
-by (nat_ind_tac "n" 1);
+by (induct_tac "n" 1);
by (Simp_tac 1);
by (Asm_simp_tac 1);
qed "sum_of_squares";
@@ -29,14 +29,14 @@
goal NatSum.thy
"Suc(Suc 2)*sum (%i. i*i*i) (Suc n) = n*n*Suc(n)*Suc(n)";
by (Simp_tac 1);
-by (nat_ind_tac "n" 1);
+by (induct_tac "n" 1);
by (Simp_tac 1);
by (Asm_simp_tac 1);
qed "sum_of_cubes";
(*The sum of the first n odd numbers equals n squared.*)
goal NatSum.thy "sum (%i. Suc(i+i)) n = n*n";
-by (nat_ind_tac "n" 1);
+by (induct_tac "n" 1);
by (Simp_tac 1);
by (Asm_simp_tac 1);
qed "sum_of_odds";