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(* Title: HOL/Integ/Lagrange.ML
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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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The following lemma essentially shows that every natural number is the sum of
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four squares, provided all prime numbers are. However, this is an abstract
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theorem about commutative rings. It has, a priori, nothing to do with nat.*)
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Goalw [Lagrange.sq_def]
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"!!x1::'a::cring. \
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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 (cring_tac 1); (*once a slow step, but now (2001) just three seconds!*)
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qed "Lagrange_lemma";
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(* A challenge by John Harrison.
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Takes forever because of the naive bottom-up strategy of the rewriter.
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Goalw [Lagrange.sq_def] "!!p1::'a::cring.\
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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 (cring_tac 1);
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*)
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