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(* Title: HOL/ex/Lagrange.thy
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ID: $Id$
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Author: Tobias Nipkow
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Copyright 1996 TU Muenchen
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This theory only contains a single theorem, which is a lemma in Lagrange's
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proof that every natural number is the sum of 4 squares. Its sole purpose is
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to demonstrate ordered rewriting for commutative rings.
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The enterprising reader might consider proving all of Lagrange's theorem.
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*)
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theory Lagrange = Main:
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constdefs sq :: "'a::times => 'a"
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"sq x == x*x"
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(* The following lemma essentially shows that every natural number is the sum
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of four squares, provided all prime numbers are. However, this is an
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abstract theorem about commutative rings. It has, a priori, nothing to do
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with nat.*)
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(*once a slow step, but now (2001) just three seconds!*)
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lemma Lagrange_lemma:
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"!!x1::'a::comm_ring_1.
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(sq x1 + sq x2 + sq x3 + sq x4) * (sq y1 + sq y2 + sq y3 + sq y4) =
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sq(x1*y1 - x2*y2 - x3*y3 - x4*y4) +
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sq(x1*y2 + x2*y1 + x3*y4 - x4*y3) +
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sq(x1*y3 - x2*y4 + x3*y1 + x4*y2) +
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sq(x1*y4 + x2*y3 - x3*y2 + x4*y1)"
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by(simp add: sq_def ring_eq_simps)
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(* A challenge by John Harrison. Takes about 4 mins on a 3GHz machine.
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lemma "!!p1::'a::comm_ring_1.
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(sq p1 + sq q1 + sq r1 + sq s1 + sq t1 + sq u1 + sq v1 + sq w1) *
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(sq p2 + sq q2 + sq r2 + sq s2 + sq t2 + sq u2 + sq v2 + sq w2)
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= sq (p1*p2 - q1*q2 - r1*r2 - s1*s2 - t1*t2 - u1*u2 - v1*v2 - w1*w2) +
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sq (p1*q2 + q1*p2 + r1*s2 - s1*r2 + t1*u2 - u1*t2 - v1*w2 + w1*v2) +
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sq (p1*r2 - q1*s2 + r1*p2 + s1*q2 + t1*v2 + u1*w2 - v1*t2 - w1*u2) +
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sq (p1*s2 + q1*r2 - r1*q2 + s1*p2 + t1*w2 - u1*v2 + v1*u2 - w1*t2) +
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sq (p1*t2 - q1*u2 - r1*v2 - s1*w2 + t1*p2 + u1*q2 + v1*r2 + w1*s2) +
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sq (p1*u2 + q1*t2 - r1*w2 + s1*v2 - t1*q2 + u1*p2 - v1*s2 + w1*r2) +
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sq (p1*v2 + q1*w2 + r1*t2 - s1*u2 - t1*r2 + u1*s2 + v1*p2 - w1*q2) +
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sq (p1*w2 - q1*v2 + r1*u2 + s1*t2 - t1*s2 - u1*r2 + v1*q2 + w1*p2)"
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by(simp add: sq_def ring_eq_simps)
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*)
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end
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