Now that 0 is overloaded, constant "zero" and its type class "zero" are
no longer needed. Also IntRingDefs is redundant
(* 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)";
*)