src/HOL/ex/Lagrange.thy
author haftmann
Wed Mar 12 19:38:14 2008 +0100 (2008-03-12)
changeset 26265 4b63b9e9b10d
parent 25475 d5a382ccb5cc
child 26480 544cef16045b
permissions -rw-r--r--
separated Random.thy from Quickcheck.thy
     1 (*  Title:      HOL/ex/Lagrange.thy
     2     ID:         $Id$
     3     Author:     Tobias Nipkow
     4     Copyright   1996 TU Muenchen
     5 *)
     6 
     7 header {* A lemma for Lagrange's theorem *}
     8 
     9 theory Lagrange imports Main begin
    10 
    11 text {* This theory only contains a single theorem, which is a lemma
    12 in Lagrange's proof that every natural number is the sum of 4 squares.
    13 Its sole purpose is to demonstrate ordered rewriting for commutative
    14 rings.
    15 
    16 The enterprising reader might consider proving all of Lagrange's
    17 theorem.  *}
    18 
    19 definition sq :: "'a::times => 'a" where "sq x == x*x"
    20 
    21 text {* The following lemma essentially shows that every natural
    22 number is the sum of four squares, provided all prime numbers are.
    23 However, this is an abstract theorem about commutative rings.  It has,
    24 a priori, nothing to do with nat. *}
    25 
    26 (* These two simprocs are even less efficient than ordered rewriting
    27    and kill the second example: *)
    28 ML_setup {*
    29   Delsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv]
    30 *}
    31 
    32 lemma Lagrange_lemma: fixes x1 :: "'a::comm_ring" shows
    33   "(sq x1 + sq x2 + sq x3 + sq x4) * (sq y1 + sq y2 + sq y3 + sq y4) =
    34    sq (x1*y1 - x2*y2 - x3*y3 - x4*y4)  +
    35    sq (x1*y2 + x2*y1 + x3*y4 - x4*y3)  +
    36    sq (x1*y3 - x2*y4 + x3*y1 + x4*y2)  +
    37    sq (x1*y4 + x2*y3 - x3*y2 + x4*y1)"
    38 by (simp add: sq_def ring_simps)
    39 
    40 
    41 text {* A challenge by John Harrison. Takes about 12s on a 1.6GHz machine. *}
    42 
    43 lemma fixes p1 :: "'a::comm_ring" shows
    44   "(sq p1 + sq q1 + sq r1 + sq s1 + sq t1 + sq u1 + sq v1 + sq w1) * 
    45    (sq p2 + sq q2 + sq r2 + sq s2 + sq t2 + sq u2 + sq v2 + sq w2) 
    46     = sq (p1*p2 - q1*q2 - r1*r2 - s1*s2 - t1*t2 - u1*u2 - v1*v2 - w1*w2) + 
    47       sq (p1*q2 + q1*p2 + r1*s2 - s1*r2 + t1*u2 - u1*t2 - v1*w2 + w1*v2) +
    48       sq (p1*r2 - q1*s2 + r1*p2 + s1*q2 + t1*v2 + u1*w2 - v1*t2 - w1*u2) +
    49       sq (p1*s2 + q1*r2 - r1*q2 + s1*p2 + t1*w2 - u1*v2 + v1*u2 - w1*t2) +
    50       sq (p1*t2 - q1*u2 - r1*v2 - s1*w2 + t1*p2 + u1*q2 + v1*r2 + w1*s2) +
    51       sq (p1*u2 + q1*t2 - r1*w2 + s1*v2 - t1*q2 + u1*p2 - v1*s2 + w1*r2) +
    52       sq (p1*v2 + q1*w2 + r1*t2 - s1*u2 - t1*r2 + u1*s2 + v1*p2 - w1*q2) +
    53       sq (p1*w2 - q1*v2 + r1*u2 + s1*t2 - t1*s2 - u1*r2 + v1*q2 + w1*p2)"
    54 by (simp add: sq_def ring_simps)
    55 
    56 end