ex/NatSum.ML
author clasohm
Wed, 02 Nov 1994 11:50:09 +0100
changeset 156 fd1be45b64bf
parent 82 0dbbf4426782
child 171 16c4ea954511
permissions -rw-r--r--
added IOA to isabelle/HOL

(*  Title: 	HOL/ex/natsum.ML
    ID:         $Id$
    Author: 	Tobias Nipkow
    Copyright   1994 TU Muenchen

Summing natural numbers, squares and cubes. Could be continued...
*)

val natsum_ss = arith_ss addsimps
  ([NatSum.sum_0,NatSum.sum_Suc] @ add_ac);

(*The sum of the first n positive integers equals n(n+1)/2.*)
goal NatSum.thy "Suc(Suc(0))*sum(%i.i,Suc(n)) = n*Suc(n)";
by (simp_tac natsum_ss 1);
by (nat_ind_tac "n" 1);
by (simp_tac natsum_ss 1);
by (asm_simp_tac natsum_ss 1);
val sum_of_naturals = result();

goal NatSum.thy
  "Suc(Suc(Suc(Suc(Suc(Suc(0))))))*sum(%i.i*i,Suc(n)) = \
\  n*Suc(n)*Suc(Suc(Suc(0))*n)";
by (simp_tac natsum_ss 1);
by (nat_ind_tac "n" 1);
by (simp_tac natsum_ss 1);
by (asm_simp_tac natsum_ss 1);
val sum_of_squares = result();

goal NatSum.thy
  "Suc(Suc(Suc(Suc(0))))*sum(%i.i*i*i,Suc(n)) = n*n*Suc(n)*Suc(n)";
by (simp_tac natsum_ss 1);
by (nat_ind_tac "n" 1);
by (simp_tac natsum_ss 1);
by (asm_simp_tac natsum_ss 1);
val sum_of_cubes = result();

(*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 (simp_tac natsum_ss 1);
by (asm_simp_tac natsum_ss 1);
val sum_of_odds = result();