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Quelle Lagrange.thy Sprache: Isabelle

(*  Title:      HOL/ex/Lagrange.thy
    Author:     Tobias Nipkow
    Copyright   1996 TU Muenchen
*)

section ‹A lemma for Lagrange's theorem›

theory Lagrange imports Main begin

text ‹This theory only contains a single theorem, which is a lemma
Lagrange's proof that every natural number is the sum of 4 squares.
sole purpose is to demonstrate ordered rewriting for commutative
.

enterprising reader might consider proving all of Lagrange's
.›

definition sq :: "'a::times => 'a" where "sq x == x*x"

text ‹The following lemma essentially shows that every natural
is the sum of four squares, provided all prime numbers are.
, this is an abstract theorem about commutative rings. It has,
priori, nothing to do with nat.›

lemma Lagrange_lemma: fixes x1 :: "'a::comm_ring" shows
  "(sq x1 + sq x2 + sq x3 + sq x4) * (sq y1 + sq y2 + sq y3 + sq y4) =
   sq (x1*y1 - x2*y2 - x3*y3 - x4*y4) +
   sq (x1*y2 + x2*y1 + x3*y4 - x4*y3) +
   sq (x1*y3 - x2*y4 + x3*y1 + x4*y2) +
   sq (x1*y4 + x2*y3 - x3*y2 + x4*y1)"
by (simp only: sq_def algebra_simps)

text ‹A challenge by John Harrison. Takes about 12s on a 1.6GHz machine.›

lemma fixes p1 :: "'a::comm_ring" shows
  "(sq p1 + sq q1 + sq r1 + sq s1 + sq t1 + sq u1 + sq v1 + sq w1) *
   (sq p2 + sq q2 + sq r2 + sq s2 + sq t2 + sq u2 + sq v2 + sq w2)
    = sq (p1*p2 - q1*q2 - r1*r2 - s1*s2 - t1*t2 - u1*u2 - v1*v2 - w1*w2) +
      sq (p1*q2 + q1*p2 + r1*s2 - s1*r2 + t1*u2 - u1*t2 - v1*w2 + w1*v2) +
      sq (p1*r2 - q1*s2 + r1*p2 + s1*q2 + t1*v2 + u1*w2 - v1*t2 - w1*u2) +
      sq (p1*s2 + q1*r2 - r1*q2 + s1*p2 + t1*w2 - u1*v2 + v1*u2 - w1*t2) +
      sq (p1*t2 - q1*u2 - r1*v2 - s1*w2 + t1*p2 + u1*q2 + v1*r2 + w1*s2) +
      sq (p1*u2 + q1*t2 - r1*w2 + s1*v2 - t1*q2 + u1*p2 - v1*s2 + w1*r2) +
      sq (p1*v2 + q1*w2 + r1*t2 - s1*u2 - t1*r2 + u1*s2 + v1*p2 - w1*q2) +
      sq (p1*w2 - q1*v2 + r1*u2 + s1*t2 - t1*s2 - u1*r2 + v1*q2 + w1*p2)"
by (simp only: sq_def algebra_simps)

end