Theory Dense_Linear_Order_Ex

Up to index of Isabelle/HOL/ex

theory Dense_Linear_Order_Ex
imports Main
begin

(*
    ID:         $Id: Dense_Linear_Order_Ex.thy,v 1.2 2007/07/22 15:53:51 chaieb Exp $
    Author:     Amine Chaieb, TU Muenchen
*)

header {* Examples for Ferrante and Rackoff's quantifier elimination procedure *}

theory Dense_Linear_Order_Ex
imports Main
begin

lemma
  "∃(y::'a::{ordered_field,recpower,number_ring, division_by_zero}) <2. x + 3* y < 0 ∧ x - y >0"
  by ferrack

lemma "~ (ALL x (y::'a::{ordered_field,recpower,number_ring, division_by_zero}). x < y --> 10*x < 11*y)"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. x < y --> (10*(x + 5*y + -1) < 60*y)"
  by ferrack

lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. x ~= y --> x < y"
  by ferrack

lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. (x ~= y & 10*x ~= 9*y & 10*x < y) --> x < y"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. (x ~= y & 5*x <= y) --> 500*x <= 100*y"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}). (EX (y::'a::{ordered_field,recpower,number_ring, division_by_zero}). 4*x + 3*y <= 0 & 4*x + 3*y >= -1)"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) < 0. (EX (y::'a::{ordered_field,recpower,number_ring, division_by_zero}) > 0. 7*x + y > 0 & x - y <= 9)"
  by ferrack

lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}). (0 < x & x < 1) --> (ALL y > 1. x + y ~= 1)"
  by ferrack

lemma "EX x. (ALL (y::'a::{ordered_field,recpower,number_ring, division_by_zero}). y < 2 -->  2*(y - x) ≤ 0 )"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}). x < 10 | x > 20 | (EX y. y>= 0 & y <= 10 & x+y = 20)"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y z. x + y < z --> y >= z --> x < 0"
  by ferrack

lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y z. x + 7*y < 5* z & 5*y >= 7*z & x < 0"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y z. abs (x + y) <= z --> (abs z = z)"
  by ferrack

lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y z. x + 7*y - 5* z < 0 & 5*y + 7*z + 3*x < 0"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y z. (abs (5*x+3*y+z) <= 5*x+3*y+z & abs (5*x+3*y+z) >= - (5*x+3*y+z)) | (abs (5*x+3*y+z) >= 5*x+3*y+z & abs (5*x+3*y+z) <= - (5*x+3*y+z))"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. x < y --> (EX z>0. x+z = y)"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. x < y --> (EX z>0. x+z = y)"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. (EX z>0. abs (x - y) <= z )"
  by ferrack

lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. (ALL z<0. (z < x --> z <= y) & (z > y --> z >= x))"
  by ferrack

lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. (ALL z>=0. abs (3*x+7*y) <= 2*z + 1)"
  by ferrack

lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. (ALL z<0. (z < x --> z <= y) & (z > y --> z >= x))"
  by ferrack

lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero})>0. (ALL y. (EX z. 13* abs z ≠ abs (12*y - x) & 5*x - 3*(abs y) <= 7*z))"
  by ferrack

lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}). abs (4*x + 17) < 4 & (ALL y . abs (x*34 - 34*y - 9) ≠ 0 --> (EX z. 5*x - 3*abs y <= 7*z))"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}). (EX y > abs (23*x - 9). (ALL z > abs (3*y - 19* abs x). x+z > 2*y))"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}). (EX y< abs (3*x - 1). (ALL z >= (3*abs x - 1). abs (12*x - 13*y + 19*z) > abs (23*x) ))"
  by ferrack

lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}). abs x < 100 & (ALL y > x. (EX z<2*y - x. 5*x - 3*y <= 7*z))"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y z w. 7*x<3*y --> 5*y < 7*z --> z < 2*w --> 7*(2*w-x) > 2*y"
  by ferrack

lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y z w. 5*x + 3*z - 17*w + abs (y - 8*x + z) <= 89"
  by ferrack

lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y z w. 5*x + 3*z - 17*w + 7* (y - 8*x + z) <= max y (7*z - x + w)"
  by ferrack

lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y z w. min (5*x + 3*z) (17*w) + 5* abs (y - 8*x + z) <= max y (7*z - x + w)"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y z. (EX w >= (x+y+z). w <= abs x + abs y + abs z)"
  by ferrack

lemma "~(ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}). (EX y z w. 3* x + z*4 = 3*y & x + y < z & x> w & 3*x < w + y))"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. (EX z w. abs (x-y) = (z-w) & z*1234 < 233*x & w ~= y)"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}). (EX y z w. min (5*x + 3*z) (17*w) + 5* abs (y - 8*x + z) <= max y (7*z - x + w))"
  by ferrack

lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y z. (ALL w >= abs (x+y+z). w >= abs x + abs y + abs z)"
  by ferrack

lemma "EX z. (ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. (EX w >= (x+y+z). w <= abs x + abs y + abs z))"
  by ferrack

lemma "EX z. (ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) < abs z. (EX y w. x< y & x < z & x> w & 3*x < w + y))"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. (EX z. (ALL w. abs (x-y) = abs (z-w) --> z < x & w ~= y))"
  by ferrack

lemma "EX y. (ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}). (EX z w. min (5*x + 3*z) (17*w) + 5* abs (y - 8*x + z) <= max y (7*z - x + w)))"
  by ferrack

lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) z. (ALL w >= 13*x - 4*z. (EX y. w >= abs x + abs y + z))"
  by ferrack

lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}). (ALL y < x. (EX z > (x+y).
  (ALL w. 5*w + 10*x - z >= y --> w + 7*x + 3*z >= 2*y)))"
  by ferrack

lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}). (ALL y. (EX z > y.
  (ALL w . w < 13 --> w + 10*x - z >= y --> 5*w + 7*x + 13*z >= 2*y)))"
  by ferrack

lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y z w. min (5*x + 3*z) (17*w) + 5* abs (y - 8*x + z) <= max y (7*z - x + w)"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}). (EX y. (ALL z>19. y <= x + z & (EX w. abs (y - x) < w)))"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}). (EX y. (ALL z>19. y <= x + z & (EX w. abs (x + z) < w - y)))"
  by ferrack

lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}). (EX y. abs y ~= abs x & (ALL z> max x y. (EX w. w ~= y & w ~= z & 3*w - z >= x + y)))"
  by ferrack

end

lemma

  y<2::'a. x + (3::'a) * y < (0::'a) ∧ (0::'a) < x - y

lemma

  ¬ (∀x y. x < y --> (10::'a) * x < (11::'a) * y)

lemma

  x y. x < y --> (10::'a) * (x + (5::'a) * y + (-1::'a)) < (60::'a) * y

lemma

  x y. x  y --> x < y

lemma

  x y. x  y ∧ (10::'a) * x  (9::'a) * y ∧ (10::'a) * x < y --> x < y

lemma

  x y. x  y ∧ (5::'a) * x  y --> (500::'a) * x  (100::'a) * y

lemma

  x. ∃y. (4::'a) * x + (3::'a) * y  (0::'a) ∧
          (-1::'a)  (4::'a) * x + (3::'a) * y

lemma

  x<0::'a. ∃y>0::'a. (0::'a) < (7::'a) * x + yx - y  (9::'a)

lemma

  x. (0::'a) < xx < (1::'a) --> (∀y>1::'a. x + y  (1::'a))

lemma

  x. ∀y<2::'a. (2::'a) * (y - x)  (0::'a)

lemma

  x. x < (10::'a) ∨ (20::'a) < x ∨ (∃y0::'a. y  (10::'a) ∧ x + y = (20::'a))

lemma

  x y z. x + y < z --> z  y --> x < (0::'a)

lemma

  x y z. x + (7::'a) * y < (5::'a) * z ∧ (7::'a) * z  (5::'a) * yx < (0::'a)

lemma

  x y z. ¦x + y¦  z --> ¦z¦ = z

lemma

  x y z.
     x + (7::'a) * y - (5::'a) * z < (0::'a) ∧
     (5::'a) * y + (7::'a) * z + (3::'a) * x < (0::'a)

lemma

  x y z.
     ¦(5::'a) * x + (3::'a) * y + z¦  (5::'a) * x + (3::'a) * y + z- ((5::'a) * x + (3::'a) * y + z)  ¦(5::'a) * x + (3::'a) * y + z¦ ∨
     (5::'a) * x + (3::'a) * y + z  ¦(5::'a) * x + (3::'a) * y + z¦¦(5::'a) * x + (3::'a) * y + z¦  - ((5::'a) * x + (3::'a) * y + z)

lemma

  x y. x < y --> (∃z>0::'a. x + z = y)

lemma

  x y. x < y --> (∃z>0::'a. x + z = y)

lemma

  x y. ∃z>0::'a. ¦x - y¦  z

lemma

  x y. ∀z<0::'a. (z < x --> z  y) ∧ (y < z --> x  z)

lemma

  x y. ∀z0::'a. ¦(3::'a) * x + (7::'a) * y¦  (2::'a) * z + (1::'a)

lemma

  x y. ∀z<0::'a. (z < x --> z  y) ∧ (y < z --> x  z)

lemma

  x>0::'a.
     ∀y. ∃z. (13::'a) * ¦z¦  ¦(12::'a) * y - x¦ ∧
             (5::'a) * x - (3::'a) * ¦y¦  (7::'a) * z

lemma

  x. ¦(4::'a) * x + (17::'a)¦ < (4::'a) ∧
      (∀y. ¦x * (34::'a) - (34::'a) * y - (9::'a)¦  (0::'a) -->
           (∃z. (5::'a) * x - (3::'a) * ¦y¦  (7::'a) * z))

lemma

  x. ∃y>¦(23::'a) * x - (9::'a)¦.
         ∀z>¦(3::'a) * y - (19::'a) * ¦x¦¦. (2::'a) * y < x + z

lemma

  x. ∃y<¦(3::'a) * x - (1::'a)¦.
         ∀z≥(3::'a) * ¦x¦ - (1::'a).
            ¦(23::'a) * x¦ < ¦(12::'a) * x - (13::'a) * y + (19::'a) * z¦

lemma

  x. ¦x¦ < (100::'a) ∧
      (∀y>x. ∃z<(2::'a) * y - x. (5::'a) * x - (3::'a) * y  (7::'a) * z)

lemma

  x y z w.
     (7::'a) * x < (3::'a) * y -->
     (5::'a) * y < (7::'a) * z -->
     z < (2::'a) * w --> (2::'a) * y < (7::'a) * ((2::'a) * w - x)

lemma

  x y z w.
     (5::'a) * x + (3::'a) * z - (17::'a) * w + ¦y - (8::'a) * x + z¦  (89::'a)

lemma

  x y z w.
     (5::'a) * x + (3::'a) * z - (17::'a) * w + (7::'a) * (y - (8::'a) * x + z)
      max y ((7::'a) * z - x + w)

lemma

  x y z w.
     min ((5::'a) * x + (3::'a) * z) ((17::'a) * w) +
     (5::'a) * ¦y - (8::'a) * x + z¦
      max y ((7::'a) * z - x + w)

lemma

  x y z. ∃wx + y + z. w  ¦x¦ + ¦y¦ + ¦z¦

lemma

  ¬ (∀x. ∃y z w.
            (3::'a) * x + z * (4::'a) = (3::'a) * yx + y < zw < x ∧ (3::'a) * x < w + y)

lemma

  x y. ∃z w. ¦x - y¦ = z - wz * (1234::'a) < (233::'a) * xw  y

lemma

  x. ∃y z w.
         min ((5::'a) * x + (3::'a) * z) ((17::'a) * w) +
         (5::'a) * ¦y - (8::'a) * x + z¦
          max y ((7::'a) * z - x + w)

lemma

  x y z. ∀w¦x + y + z¦. ¦x¦ + ¦y¦ + ¦z¦  w

lemma

  z. ∀x y. ∃wx + y + z. w  ¦x¦ + ¦y¦ + ¦z¦

lemma

  z. ∀x<¦z¦. ∃y w. x < yx < zw < x ∧ (3::'a) * x < w + y

lemma

  x y. ∃z. ∀w. ¦x - y¦ = ¦z - w¦ --> z < xw  y

lemma

  y. ∀x. ∃z w. min ((5::'a) * x + (3::'a) * z) ((17::'a) * w) +
                (5::'a) * ¦y - (8::'a) * x + z¦
                 max y ((7::'a) * z - x + w)

lemma

  x z. ∀w≥(13::'a) * x - (4::'a) * z. ∃y. ¦x¦ + ¦y¦ + z  w

lemma

  x. ∀y<x. ∃z>x + y.
               ∀w. y  (5::'a) * w + (10::'a) * x - z -->
                   (2::'a) * y  w + (7::'a) * x + (3::'a) * z

lemma

  x. ∀y. ∃z>y. ∀w<13::'a.
                   y  w + (10::'a) * x - z -->
                   (2::'a) * y  (5::'a) * w + (7::'a) * x + (13::'a) * z

lemma

  x y z w.
     min ((5::'a) * x + (3::'a) * z) ((17::'a) * w) +
     (5::'a) * ¦y - (8::'a) * x + z¦
      max y ((7::'a) * z - x + w)

lemma

  x. ∃y. ∀z>19::'a. y  x + z ∧ (∃w. ¦y - x¦ < w)

lemma

  x. ∃y. ∀z>19::'a. y  x + z ∧ (∃w. ¦x + z¦ < w - y)

lemma

  x. ∃y. ¦y¦  ¦x¦ ∧ (∀z>max x y. ∃w. w  yw  zx + y  (3::'a) * w - z)