src/HOL/ex/Lagrange.thy
author wenzelm
Sun Nov 02 18:21:45 2014 +0100 (2014-11-02)
changeset 58889 5b7a9633cfa8
parent 58776 95e58e04e534
child 61343 5b5656a63bd6
permissions -rw-r--r--
modernized header uniformly as section;
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(*  Title:      HOL/ex/Lagrange.thy
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    Author:     Tobias Nipkow
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    Copyright   1996 TU Muenchen
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*)
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section {* A lemma for Lagrange's theorem *}
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theory Lagrange imports Main begin
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text {* This theory only contains a single theorem, which is a lemma
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in Lagrange's proof that every natural number is the sum of 4 squares.
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Its sole purpose is to demonstrate ordered rewriting for commutative
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rings.
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The enterprising reader might consider proving all of Lagrange's
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theorem.  *}
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definition sq :: "'a::times => 'a" where "sq x == x*x"
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text {* The following lemma essentially shows that every natural
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number is the sum of four squares, provided all prime numbers are.
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However, this is an abstract theorem about commutative rings.  It has,
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a priori, nothing to do with nat. *}
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lemma Lagrange_lemma: fixes x1 :: "'a::comm_ring" shows
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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 only: sq_def algebra_simps)
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text {* A challenge by John Harrison. Takes about 12s on a 1.6GHz machine. *}
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lemma fixes p1 :: "'a::comm_ring" shows
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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 only: sq_def algebra_simps)
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end