--- a/src/HOL/ex/NatSum.ML Mon May 08 16:59:02 2000 +0200
+++ b/src/HOL/ex/NatSum.ML Mon May 08 16:59:18 2000 +0200
@@ -3,7 +3,7 @@
Author: Tobias Nipkow
Copyright 1994 TU Muenchen
-Summing natural numbers, squares and cubes. Could be continued...
+Summing natural numbers, squares, cubes, etc.
Originally demonstrated permutative rewriting, but add_ac is no longer needed
thanks to new simprocs.
@@ -32,3 +32,32 @@
by (induct_tac "n" 1);
by Auto_tac;
qed "sum_of_cubes";
+
+(** Sum of fourth powers requires lemmas **)
+
+Goal "[| #1 <= (j::nat); k <= m |] ==> k <= j*m";
+by (dtac mult_le_mono 1);
+by Auto_tac;
+qed "mult_le_1_mono";
+
+Addsimps [diff_mult_distrib, diff_mult_distrib2];
+
+(*Thanks to Sloane's On-Line Encyclopedia of Integer Sequences,
+ http://www.research.att.com/~njas/sequences/*)
+Goal "#30 * sum (%i. i*i*i*i) (Suc n) = \
+\ n * Suc(n) * Suc(#2*n) * (#3*n*n + #3*n - #1)";
+by (induct_tac "n" 1);
+by (Simp_tac 1);
+(*Automating this inequality proof would make the proof trivial*)
+by (subgoal_tac "n <= #10 * (n * (n * n)) + (#15 * (n * (n * (n * n))) + \
+\ #6 * (n * (n * (n * (n * n)))))" 1);
+(*In simplifying we want only the outer Suc argument to be unfolded.
+ Thus the result matches the induction hypothesis (also with Suc). *)
+by (asm_simp_tac (simpset() delsimps [sum_Suc]
+ addsimps [inst "n" "Suc ?m" sum_Suc]) 1);
+by (rtac ([mult_le_1_mono, le_add1] MRS le_trans) 1);
+by (rtac le_cube 2);
+by (Simp_tac 1);
+qed "sum_of_fourth_powers";
+
+--- a/src/HOL/ex/NatSum.thy Mon May 08 16:59:02 2000 +0200
+++ b/src/HOL/ex/NatSum.thy Mon May 08 16:59:18 2000 +0200
@@ -1,4 +1,4 @@
-(* Title: HOL/ex/natsum.thy
+(* Title: HOL/ex/NatSum.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1994 TU Muenchen
@@ -9,7 +9,7 @@
NatSum = Main +
consts sum :: [nat=>nat, nat] => nat
primrec
- "sum f 0 = 0"
- "sum f (Suc n) = f(n) + sum f n"
+ sum_0 "sum f 0 = 0"
+ sum_Suc "sum f (Suc n) = f(n) + sum f n"
end