src/HOL/ex/Lagrange.thy
author wenzelm
Wed, 27 Jan 2021 13:08:07 +0100
changeset 73185 b310b93563f6
parent 61343 5b5656a63bd6
permissions -rw-r--r--
more robust: support other_isabelle.init_settings for build_history before b93404a4c3dd;

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

section \<open>A lemma for Lagrange's theorem\<close>

theory Lagrange imports Main begin

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

The enterprising reader might consider proving all of Lagrange's
theorem.\<close>

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

text \<open>The following lemma essentially shows that every natural
number is the sum of four squares, provided all prime numbers are.
However, this is an abstract theorem about commutative rings.  It has,
a priori, nothing to do with nat.\<close>

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 \<open>A challenge by John Harrison. Takes about 12s on a 1.6GHz machine.\<close>

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