src/HOL/ex/Lagrange.ML
changeset 5078 7b5ea59c0275
child 11375 a6730c90e753
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Lagrange.ML	Thu Jun 25 13:57:34 1998 +0200
@@ -0,0 +1,37 @@
+(*  Title:      HOL/Integ/Lagrange.ML
+    ID:         $Id$
+    Author:     Tobias Nipkow
+    Copyright   1996 TU Muenchen
+
+
+The following lemma essentially shows that all composite natural numbers are
+sums of fours squares, provided all prime numbers are. However, this is an
+abstract thm about commutative rings and has a priori nothing to do with nat.
+*)
+
+Goalw [Lagrange.sq_def] "!!x1::'a::cring. \
+\  (sq x1 + sq x2 + sq x3 + sq x4) * (sq y1 + sq y2 + sq y3 + sq y4) = \
+\  sq(x1*y1 - x2*y2 - x3*y3 - x4*y4)  + \
+\  sq(x1*y2 + x2*y1 + x3*y4 - x4*y3)  + \
+\  sq(x1*y3 - x2*y4 + x3*y1 + x4*y2)  + \
+\  sq(x1*y4 + x2*y3 - x3*y2 + x4*y1)";
+(*Takes up to three minutes...*)
+by (cring_tac 1);
+qed "Lagrange_lemma";
+
+(* A challenge by John Harrison.
+   Takes forever because of the naive bottom-up strategy of the rewriter.
+
+Goalw [Lagrange.sq_def] "!!p1::'a::cring.\
+\ (sq p1 + sq q1 + sq r1 + sq s1 + sq t1 + sq u1 + sq v1 + sq w1) * \
+\ (sq p2 + sq q2 + sq r2 + sq s2 + sq t2 + sq u2 + sq v2 + sq w2) \
+\  = sq (p1*p2 - q1*q2 - r1*r2 - s1*s2 - t1*t2 - u1*u2 - v1*v2 - w1*w2) + \
+\    sq (p1*q2 + q1*p2 + r1*s2 - s1*r2 + t1*u2 - u1*t2 - v1*w2 + w1*v2) +\
+\    sq (p1*r2 - q1*s2 + r1*p2 + s1*q2 + t1*v2 + u1*w2 - v1*t2 - w1*u2) +\
+\    sq (p1*s2 + q1*r2 - r1*q2 + s1*p2 + t1*w2 - u1*v2 + v1*u2 - w1*t2) +\
+\    sq (p1*t2 - q1*u2 - r1*v2 - s1*w2 + t1*p2 + u1*q2 + v1*r2 + w1*s2) +\
+\    sq (p1*u2 + q1*t2 - r1*w2 + s1*v2 - t1*q2 + u1*p2 - v1*s2 + w1*r2) +\
+\    sq (p1*v2 + q1*w2 + r1*t2 - s1*u2 - t1*r2 + u1*s2 + v1*p2 - w1*q2) +\
+\    sq (p1*w2 - q1*v2 + r1*u2 + s1*t2 - t1*s2 - u1*r2 + v1*q2 + w1*p2)";
+
+*)