src/HOL/ex/Lagrange.thy
author obua
Mon Apr 10 16:00:34 2006 +0200 (2006-04-10)
changeset 19404 9bf2cdc9e8e8
parent 17388 495c799df31d
child 19736 d8d0f8f51d69
permissions -rw-r--r--
Moved stuff from Ring_and_Field to Matrix
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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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*)
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header {* 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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constdefs sq :: "'a::times => 'a"
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         "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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ML"Delsimprocs[ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv]"
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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.
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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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text{*A challenge by John Harrison. Takes about 74s on a 2.5GHz Apple G5.*}
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lemma "!!p1::'a::comm_ring.
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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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oops
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(*
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by(simp add: sq_def ring_eq_simps)
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*)
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end