Groebner Basis Examples.

--- /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 \<Longrightarrow> x = z + 3 \<Longrightarrow> 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 \<longleftrightarrow> (x = 3 \<or> x = -1)"
+  by algebra
+
+theorem "x* (x\<twosuperior> - x  - 5) - 3 = (0::int) \<longleftrightarrow> (x = 3 \<or> 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 \<times> int"
+
+definition collinear ::"point \<Rightarrow> point \<Rightarrow> point \<Rightarrow> bool" where
+  "collinear \<equiv> \<lambda>(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\<twosuperior> + s\<twosuperior> = 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