src/HOL/ex/Lagrange.thy
author haftmann
Fri Oct 10 19:55:32 2014 +0200 (2014-10-10)
changeset 58646 cd63a4b12a33
parent 37885 c43805c80eb6
child 58776 95e58e04e534
permissions -rw-r--r--
specialized specification: avoid trivial instances
     1 (*  Title:      HOL/ex/Lagrange.thy
     2     Author:     Tobias Nipkow
     3     Copyright   1996 TU Muenchen
     4 *)
     5 
     6 header {* A lemma for Lagrange's theorem *}
     7 
     8 theory Lagrange imports Main begin
     9 
    10 text {* This theory only contains a single theorem, which is a lemma
    11 in Lagrange's proof that every natural number is the sum of 4 squares.
    12 Its sole purpose is to demonstrate ordered rewriting for commutative
    13 rings.
    14 
    15 The enterprising reader might consider proving all of Lagrange's
    16 theorem.  *}
    17 
    18 definition sq :: "'a::times => 'a" where "sq x == x*x"
    19 
    20 text {* The following lemma essentially shows that every natural
    21 number is the sum of four squares, provided all prime numbers are.
    22 However, this is an abstract theorem about commutative rings.  It has,
    23 a priori, nothing to do with nat. *}
    24 
    25 lemma Lagrange_lemma: fixes x1 :: "'a::comm_ring" shows
    26   "(sq x1 + sq x2 + sq x3 + sq x4) * (sq y1 + sq y2 + sq y3 + sq y4) =
    27    sq (x1*y1 - x2*y2 - x3*y3 - x4*y4)  +
    28    sq (x1*y2 + x2*y1 + x3*y4 - x4*y3)  +
    29    sq (x1*y3 - x2*y4 + x3*y1 + x4*y2)  +
    30    sq (x1*y4 + x2*y3 - x3*y2 + x4*y1)"
    31 by (simp only: sq_def field_simps)
    32 
    33 
    34 text {* A challenge by John Harrison. Takes about 12s on a 1.6GHz machine. *}
    35 
    36 lemma fixes p1 :: "'a::comm_ring" shows
    37   "(sq p1 + sq q1 + sq r1 + sq s1 + sq t1 + sq u1 + sq v1 + sq w1) * 
    38    (sq p2 + sq q2 + sq r2 + sq s2 + sq t2 + sq u2 + sq v2 + sq w2) 
    39     = sq (p1*p2 - q1*q2 - r1*r2 - s1*s2 - t1*t2 - u1*u2 - v1*v2 - w1*w2) + 
    40       sq (p1*q2 + q1*p2 + r1*s2 - s1*r2 + t1*u2 - u1*t2 - v1*w2 + w1*v2) +
    41       sq (p1*r2 - q1*s2 + r1*p2 + s1*q2 + t1*v2 + u1*w2 - v1*t2 - w1*u2) +
    42       sq (p1*s2 + q1*r2 - r1*q2 + s1*p2 + t1*w2 - u1*v2 + v1*u2 - w1*t2) +
    43       sq (p1*t2 - q1*u2 - r1*v2 - s1*w2 + t1*p2 + u1*q2 + v1*r2 + w1*s2) +
    44       sq (p1*u2 + q1*t2 - r1*w2 + s1*v2 - t1*q2 + u1*p2 - v1*s2 + w1*r2) +
    45       sq (p1*v2 + q1*w2 + r1*t2 - s1*u2 - t1*r2 + u1*s2 + v1*p2 - w1*q2) +
    46       sq (p1*w2 - q1*v2 + r1*u2 + s1*t2 - t1*s2 - u1*r2 + v1*q2 + w1*p2)"
    47 by (simp only: sq_def field_simps)
    48 
    49 end