diff -r 25c84ca5e9a9 -r c6d5ab154c7c src/HOL/ex/Groebner_Examples.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/ex/Groebner_Examples.thy Tue Jun 05 22:47:49 2007 +0200 @@ -0,0 +1,97 @@ +(* Title: HOL/ex/Groebner_Examples.thy + ID: \$Id\$ + Author: Amine Chaieb, TU Muenchen +*) + +header {* Groebner Basis Examples *} + +theory Groebner_Examples +imports Main +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* (x\ - x - 5) - 3 = (0::int) \ (x = 3 \ x = -1)" + 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)" + unfolding sq_def by algebra + +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)" + unfolding sq_def by algebra + + +subsection {* Colinearity is invariant by rotation *} + +types point = "int \ int" + +definition collinear ::"point \ point \ point \ bool" where + "collinear \ \(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 "c\ + s\ = 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 unfolding collinear_def split_def fst_conv snd_conv + by algebra + +end