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theory Groebner_Examples(* Title: HOL/ex/Groebner_Examples.thy ID: $Id: Groebner_Examples.thy,v 1.5 2007/10/31 11:19:45 chaieb Exp $ Author: Amine Chaieb, TU Muenchen *) header {* Groebner Basis Examples *} theory Groebner_Examples imports Groebner_Basis begin subsection {* Basic examples *} lemma "3 ^ 3 == (?X::'a::{number_ring,recpower})" by sring_norm lemma "(x - (-2))^5 == ?X::int" by sring_norm lemma "(x - (-2))^5 * (y - 78) ^ 8 == ?X::int" by sring_norm lemma "((-3) ^ (Suc (Suc (Suc 0)))) == (X::'a::{number_ring,recpower})" apply (simp only: power_Suc power_0) apply (simp only: comp_arith) oops lemma "((x::int) + y)^3 - 1 = (x - z)^2 - 10 ==> x = z + 3 ==> x = - y" by algebra lemma "(4::nat) + 4 = 3 + 5" by algebra lemma "(4::int) + 0 = 4" apply algebra? by simp lemma assumes "a * x^2 + b * x + c = (0::int)" and "d * x^2 + e * x + f = 0" shows "d^2*c^2 - 2*d*c*a*f + a^2*f^2 - e*d*b*c - e*b*a*f + a*e^2*c + f*d*b^2 = 0" using assms by algebra lemma "(x::int)^3 - x^2 - 5*x - 3 = 0 <-> (x = 3 ∨ x = -1)" by algebra theorem "x* (x² - x - 5) - 3 = (0::int) <-> (x = 3 ∨ x = -1)" by algebra lemma fixes x::"'a::{idom,recpower,number_ring}" shows "x^2*y = x^2 & x*y^2 = y^2 <-> x=1 & y=1 | x=0 & y=0" by algebra subsection {* Lemmas for Lagrange's theorem *} definition sq :: "'a::times => 'a" where "sq x == x*x" lemma fixes x1 :: "'a::{idom,recpower,number_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 (algebra add: sq_def) lemma fixes p1 :: "'a::{idom,recpower,number_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 (algebra add: sq_def) subsection {* Colinearity is invariant by rotation *} types point = "int × int" definition collinear ::"point => point => point => bool" where "collinear ≡ λ(Ax,Ay) (Bx,By) (Cx,Cy). ((Ax - Bx) * (By - Cy) = (Ay - By) * (Bx - Cx))" lemma collinear_inv_rotation: assumes "collinear (Ax, Ay) (Bx, By) (Cx, Cy)" and "c² + s² = 1" shows "collinear (Ax * c - Ay * s, Ay * c + Ax * s) (Bx * c - By * s, By * c + Bx * s) (Cx * c - Cy * s, Cy * c + Cx * s)" using assms by (algebra add: collinear_def split_def fst_conv snd_conv) lemma "EX (d::int). a*y - a*x = n*d ==> EX u v. a*u + n*v = 1 ==> EX e. y - x = n*e" apply algebra done end
lemma
(3::'a) ^ 3 == 27::'a
lemma
(x - -2) ^ 5 ==
x ^ 5 + (10 * x ^ 4 + (40 * x ^ 3 + (80 * x ^ 2 + (80 * x + 32))))
lemma
(x - -2) ^ 5 * (y - 78) ^ 8 ==
x ^ 5 * y ^ 8 +
(-624 * (x ^ 5 * y ^ 7) +
(10 * (x ^ 4 * y ^ 8) +
(170352 * (x ^ 5 * y ^ 6) +
(-6240 * (x ^ 4 * y ^ 7) +
(40 * (x ^ 3 * y ^ 8) +
(-26574912 * (x ^ 5 * y ^ 5) +
(1703520 * (x ^ 4 * y ^ 6) +
(-24960 * (x ^ 3 * y ^ 7) +
(80 * (x ^ 2 * y ^ 8) +
(2591053920 * (x ^ 5 * y ^ 4) +
(-265749120 * (x ^ 4 * y ^ 5) +
(6814080 * (x ^ 3 * y ^ 6) +
(-49920 * (x ^ 2 * y ^ 7) +
(80 * (x * y ^ 8) +
(-161681764608 * (x ^ 5 * y ^ 3) +
(25910539200 * (x ^ 4 * y ^ 4) +
(-1062996480 * (x ^ 3 * y ^ 5) +
(13628160 * (x ^ 2 * y ^ 6) +
(-49920 * (x * y ^ 7) +
(32 * y ^ 8 +
(6305588819712 * (x ^ 5 * y ^ 2) +
(-1616817646080 * (x ^ 4 * y ^ 3) +
(103642156800 * (x ^ 3 * y ^ 4) +
(-2125992960 * (x ^ 2 * y ^ 5) +
(13628160 * (x * y ^ 6) +
(-19968 * y ^ 7 +
(-140524550839296 * (x ^ 5 * y) +
(63055888197120 * (x ^ 4 * y ^ 2) +
(-6467270584320 * (x ^ 3 * y ^ 3) +
(207284313600 * (x ^ 2 * y ^ 4) +
(-2125992960 * (x * y ^ 5) +
(5451264 * y ^ 6 +
(1370114370683136 * x ^ 5 +
(-1405245508392960 * (x ^ 4 * y) +
(252223552788480 * (x ^ 3 * y ^ 2) +
(-12934541168640 * (x ^ 2 * y ^ 3) +
(207284313600 * (x * y ^ 4) +
(-850397184 * y ^ 5 +
(13701143706831360 * x ^ 4 +
(-5620982033571840 * (x ^ 3 * y) +
(504447105576960 * (x ^ 2 * y ^ 2) +
(-12934541168640 * (x * y ^ 3) +
(82913725440 * y ^ 4 +
(54804574827325440 * x ^ 3 +
(-11241964067143680 * (x ^ 2 * y) +
(504447105576960 * (x * y ^ 2) +
(-5173816467456 * y ^ 3 +
(109609149654650880 * x ^ 2 +
(-11241964067143680 * (x * y) +
(201778842230784 * y ^ 2 +
(109609149654650880 * x +
(-4496785626857472 * y +
43843659861860352))))))))))))))))))))))))))))))))))))))))))))))))))))
lemma
[| (x + y) ^ 3 - 1 = (x - z) ^ 2 - 10; x = z + 3 |] ==> x = - y
lemma
4 + 4 = 3 + 5
lemma
4 + 0 = 4
lemma
[| a * x ^ 2 + b * x + c = 0; d * x ^ 2 + e * x + f = 0 |]
==> d ^ 2 * c ^ 2 - 2 * d * c * a * f + a ^ 2 * f ^ 2 - e * d * b * c -
e * b * a * f +
a * e ^ 2 * c +
f * d * b ^ 2 =
0
lemma
(x ^ 3 - x ^ 2 - 5 * x - 3 = 0) = (x = 3 ∨ x = -1)
theorem
(x * (x ^ 2 - x - 5) - 3 = 0) = (x = 3 ∨ x = -1)
lemma
(x ^ 2 * y = x ^ 2 ∧ x * y ^ 2 = y ^ 2) =
(x = (1::'a) ∧ y = (1::'a) ∨ x = (0::'a) ∧ y = (0::'a))
lemma
(sq x1.0 + sq x2.0 + sq x3.0 + sq x4.0) *
(sq y1.0 + sq y2.0 + sq y3.0 + sq y4.0) =
sq (x1.0 * y1.0 - x2.0 * y2.0 - x3.0 * y3.0 - x4.0 * y4.0) +
sq (x1.0 * y2.0 + x2.0 * y1.0 + x3.0 * y4.0 - x4.0 * y3.0) +
sq (x1.0 * y3.0 - x2.0 * y4.0 + x3.0 * y1.0 + x4.0 * y2.0) +
sq (x1.0 * y4.0 + x2.0 * y3.0 - x3.0 * y2.0 + x4.0 * y1.0)
lemma
(sq p1.0 + sq q1.0 + sq r1.0 + sq s1.0 + sq t1.0 + sq u1.0 + sq v1.0 +
sq w1.0) *
(sq p2.0 + sq q2.0 + sq r2.0 + sq s2.0 + sq t2.0 + sq u2.0 + sq v2.0 +
sq w2.0) =
sq (p1.0 * p2.0 - q1.0 * q2.0 - r1.0 * r2.0 - s1.0 * s2.0 - t1.0 * t2.0 -
u1.0 * u2.0 -
v1.0 * v2.0 -
w1.0 * w2.0) +
sq (p1.0 * q2.0 + q1.0 * p2.0 + r1.0 * s2.0 - s1.0 * r2.0 + t1.0 * u2.0 -
u1.0 * t2.0 -
v1.0 * w2.0 +
w1.0 * v2.0) +
sq (p1.0 * r2.0 - q1.0 * s2.0 + r1.0 * p2.0 + s1.0 * q2.0 + t1.0 * v2.0 +
u1.0 * w2.0 -
v1.0 * t2.0 -
w1.0 * u2.0) +
sq (p1.0 * s2.0 + q1.0 * r2.0 - r1.0 * q2.0 + s1.0 * p2.0 + t1.0 * w2.0 -
u1.0 * v2.0 +
v1.0 * u2.0 -
w1.0 * t2.0) +
sq (p1.0 * t2.0 - q1.0 * u2.0 - r1.0 * v2.0 - s1.0 * w2.0 + t1.0 * p2.0 +
u1.0 * q2.0 +
v1.0 * r2.0 +
w1.0 * s2.0) +
sq (p1.0 * u2.0 + q1.0 * t2.0 - r1.0 * w2.0 + s1.0 * v2.0 - t1.0 * q2.0 +
u1.0 * p2.0 -
v1.0 * s2.0 +
w1.0 * r2.0) +
sq (p1.0 * v2.0 + q1.0 * w2.0 + r1.0 * t2.0 - s1.0 * u2.0 - t1.0 * r2.0 +
u1.0 * s2.0 +
v1.0 * p2.0 -
w1.0 * q2.0) +
sq (p1.0 * w2.0 - q1.0 * v2.0 + r1.0 * u2.0 + s1.0 * t2.0 - t1.0 * s2.0 -
u1.0 * r2.0 +
v1.0 * q2.0 +
w1.0 * p2.0)
lemma collinear_inv_rotation:
[| collinear (Ax, Ay) (Bx, By) (Cx, Cy); c ^ 2 + s ^ 2 = 1 |]
==> collinear (Ax * c - Ay * s, Ay * c + Ax * s)
(Bx * c - By * s, By * c + Bx * s) (Cx * c - Cy * s, Cy * c + Cx * s)
lemma
[| ∃d. a * y - a * x = n * d; ∃u v. a * u + n * v = 1 |] ==> ∃e. y - x = n * e