Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
(* Title: HOL/ex/Lagrange.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1996 TU Muenchen
*)
header {* A lemma for Lagrange's theorem *}
theory Lagrange imports Main begin
text {* This theory only contains a single theorem, which is a lemma
in Lagrange's proof that every natural number is the sum of 4 squares.
Its sole purpose is to demonstrate ordered rewriting for commutative
rings.
The enterprising reader might consider proving all of Lagrange's
theorem. *}
definition sq :: "'a::times => 'a" where "sq x == x*x"
text {* The following lemma essentially shows that every natural
number is the sum of four squares, provided all prime numbers are.
However, this is an abstract theorem about commutative rings. It has,
a priori, nothing to do with nat. *}
(* These two simprocs are even less efficient than ordered rewriting
and kill the second example: *)
ML_setup {*
Delsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv]
*}
lemma Lagrange_lemma: fixes x1 :: "'a::comm_ring" shows
"(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)"
by (simp add: sq_def ring_simps)
text {* A challenge by John Harrison. Takes about 12s on a 1.6GHz machine. *}
lemma fixes p1 :: "'a::comm_ring" shows
"(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)"
by (simp add: sq_def ring_simps)
end