src/HOL/Decision_Procs/ex/Dense_Linear_Order_Ex.thy
author nipkow
Tue Feb 23 16:25:08 2016 +0100 (2016-02-23)
changeset 62390 842917225d56
parent 60540 b7b71952c194
permissions -rw-r--r--
more canonical names
     1 (* Author:     Amine Chaieb, TU Muenchen *)
     2 
     3 section \<open>Examples for Ferrante and Rackoff's quantifier elimination procedure\<close>
     4 
     5 theory Dense_Linear_Order_Ex
     6 imports "../Dense_Linear_Order"
     7 begin
     8 
     9 lemma "\<exists>(y::'a::linordered_field) < 2. x + 3* y < 0 \<and> x - y > 0"
    10   by ferrack
    11 
    12 lemma "\<not> (\<forall>x (y::'a::linordered_field). x < y \<longrightarrow> 10 * x < 11 * y)"
    13   by ferrack
    14 
    15 lemma "\<forall>(x::'a::linordered_field) y. x < y \<longrightarrow> 10 * (x + 5 * y + -1) < 60 * y"
    16   by ferrack
    17 
    18 lemma "\<exists>(x::'a::linordered_field) y. x \<noteq> y \<longrightarrow> x < y"
    19   by ferrack
    20 
    21 lemma "\<exists>(x::'a::linordered_field) y. x \<noteq> y \<and> 10 * x \<noteq> 9 * y \<and> 10 * x < y \<longrightarrow> x < y"
    22   by ferrack
    23 
    24 lemma "\<forall>(x::'a::linordered_field) y. x \<noteq> y \<and> 5 * x \<le> y \<longrightarrow> 500 * x \<le> 100 * y"
    25   by ferrack
    26 
    27 lemma "\<forall>x::'a::linordered_field. \<exists>y::'a::linordered_field. 4 * x + 3 * y \<le> 0 \<and> 4 * x + 3 * y \<ge> -1"
    28   by ferrack
    29 
    30 lemma "\<forall>(x::'a::linordered_field) < 0. \<exists>(y::'a::linordered_field) > 0. 7 * x + y > 0 \<and> x - y \<le> 9"
    31   by ferrack
    32 
    33 lemma "\<exists>x::'a::linordered_field. 0 < x \<and> x < 1 \<longrightarrow> (\<forall>y > 1. x + y \<noteq> 1)"
    34   by ferrack
    35 
    36 lemma "\<exists>x. \<forall>y::'a::linordered_field. y < 2 \<longrightarrow> 2 * (y - x) \<le> 0"
    37   by ferrack
    38 
    39 lemma "\<forall>x::'a::linordered_field. x < 10 \<or> x > 20 \<or> (\<exists>y. y \<ge> 0 \<and> y \<le> 10 \<and> x + y = 20)"
    40   by ferrack
    41 
    42 lemma "\<forall>(x::'a::linordered_field) y z. x + y < z \<longrightarrow> y \<ge> z \<longrightarrow> x < 0"
    43   by ferrack
    44 
    45 lemma "\<exists>(x::'a::linordered_field) y z. x + 7 * y < 5 * z \<and> 5 * y \<ge> 7 * z \<and> x < 0"
    46   by ferrack
    47 
    48 lemma "\<forall>(x::'a::linordered_field) y z. \<bar>x + y\<bar> \<le> z \<longrightarrow> \<bar>z\<bar> = z"
    49   by ferrack
    50 
    51 lemma "\<exists>(x::'a::linordered_field) y z. x + 7 * y - 5 * z < 0 \<and> 5 * y + 7 * z + 3 * x < 0"
    52   by ferrack
    53 
    54 lemma "\<forall>(x::'a::linordered_field) y z.
    55   (\<bar>5 * x + 3 * y + z\<bar> \<le> 5 * x + 3 * y + z \<and> \<bar>5 * x + 3 * y + z\<bar> \<ge> - (5 * x + 3 * y + z)) \<or>
    56   (\<bar>5 * x + 3 * y + z\<bar> \<ge> 5 * x + 3 * y + z \<and> \<bar>5 * x + 3 * y + z\<bar> \<le> - (5 * x + 3 * y + z))"
    57   by ferrack
    58 
    59 lemma "\<forall>(x::'a::linordered_field) y. x < y \<longrightarrow> (\<exists>z>0. x + z = y)"
    60   by ferrack
    61 
    62 lemma "\<forall>(x::'a::linordered_field) y. x < y \<longrightarrow> (\<exists>z>0. x + z = y)"
    63   by ferrack
    64 
    65 lemma "\<forall>(x::'a::linordered_field) y. \<exists>z>0. \<bar>x - y\<bar> \<le> z"
    66   by ferrack
    67 
    68 lemma "\<exists>(x::'a::linordered_field) y. \<forall>z<0. (z < x \<longrightarrow> z \<le> y) \<and> (z > y \<longrightarrow> z \<ge> x)"
    69   by ferrack
    70 
    71 lemma "\<exists>(x::'a::linordered_field) y. \<forall>z\<ge>0. \<bar>3 * x + 7 * y\<bar> \<le> 2 * z + 1"
    72   by ferrack
    73 
    74 lemma "\<exists>(x::'a::linordered_field) y. \<forall>z<0. (z < x \<longrightarrow> z \<le> y) \<and> (z > y \<longrightarrow> z \<ge> x)"
    75   by ferrack
    76 
    77 lemma "\<exists>(x::'a::linordered_field) > 0. \<forall>y. \<exists>z. 13 * \<bar>z\<bar> \<noteq> \<bar>12 * y - x\<bar> \<and> 5 * x - 3 * \<bar>y\<bar> \<le> 7 * z"
    78   by ferrack
    79 
    80 lemma "\<exists>x::'a::linordered_field.
    81   \<bar>4 * x + 17\<bar> < 4 \<and> (\<forall>y. \<bar>x * 34 - 34 * y - 9\<bar> \<noteq> 0 \<longrightarrow> (\<exists>z. 5 * x - 3 * \<bar>y\<bar> \<le> 7 * z))"
    82   by ferrack
    83 
    84 lemma "\<forall>x::'a::linordered_field. \<exists>y > \<bar>23 * x - 9\<bar>. \<forall>z > \<bar>3 * y - 19 * \<bar>x\<bar>\<bar>. x + z > 2 * y"
    85   by ferrack
    86 
    87 lemma "\<forall>x::'a::linordered_field.
    88   \<exists>y < \<bar>3 * x - 1\<bar>. \<forall>z \<ge> 3 * \<bar>x\<bar> - 1. \<bar>12 * x - 13 * y + 19 * z\<bar> > \<bar>23 * x\<bar>"
    89   by ferrack
    90 
    91 lemma "\<exists>x::'a::linordered_field. \<bar>x\<bar> < 100 \<and> (\<forall>y > x. (\<exists>z < 2 * y - x. 5 * x - 3 * y \<le> 7 * z))"
    92   by ferrack
    93 
    94 lemma "\<forall>(x::'a::linordered_field) y z w.
    95   7 * x < 3 * y \<longrightarrow> 5 * y < 7 * z \<longrightarrow> z < 2 * w \<longrightarrow> 7 * (2 * w - x) > 2 * y"
    96   by ferrack
    97 
    98 lemma "\<exists>(x::'a::linordered_field) y z w. 5 * x + 3 * z - 17 * w + \<bar>y - 8 * x + z\<bar> \<le> 89"
    99   by ferrack
   100 
   101 lemma "\<exists>(x::'a::linordered_field) y z w.
   102   5 * x + 3 * z - 17 * w + 7 * (y - 8 * x + z) \<le> max y (7 * z - x + w)"
   103   by ferrack
   104 
   105 lemma "\<exists>(x::'a::linordered_field) y z w.
   106   min (5 * x + 3 * z) (17 * w) + 5 * \<bar>y - 8 * x + z\<bar> \<le> max y (7 * z - x + w)"
   107   by ferrack
   108 
   109 lemma "\<forall>(x::'a::linordered_field) y z. \<exists>w \<ge> x + y + z. w \<le> \<bar>x\<bar> + \<bar>y\<bar> + \<bar>z\<bar>"
   110   by ferrack
   111 
   112 lemma "\<not> (\<forall>x::'a::linordered_field. \<exists>y z w.
   113   3 * x + z * 4 = 3 * y \<and> x + y < z \<and> x > w \<and> 3 * x < w + y)"
   114   by ferrack
   115 
   116 lemma "\<forall>(x::'a::linordered_field) y. \<exists>z w. \<bar>x - y\<bar> = z - w \<and> z * 1234 < 233 * x \<and> w \<noteq> y"
   117   by ferrack
   118 
   119 lemma "\<forall>x::'a::linordered_field. \<exists>y z w.
   120   min (5 * x + 3 * z) (17 * w) + 5 * \<bar>y - 8 * x + z\<bar> \<le> max y (7 * z - x + w)"
   121   by ferrack
   122 
   123 lemma "\<exists>(x::'a::linordered_field) y z. \<forall>w \<ge> \<bar>x + y + z\<bar>. w \<ge> \<bar>x\<bar> + \<bar>y\<bar> + \<bar>z\<bar>"
   124   by ferrack
   125 
   126 lemma "\<exists>z. \<forall>(x::'a::linordered_field) y. \<exists>w \<ge> x + y + z. w \<le> \<bar>x\<bar> + \<bar>y\<bar> + \<bar>z\<bar>"
   127   by ferrack
   128 
   129 lemma "\<exists>z. \<forall>(x::'a::linordered_field) < \<bar>z\<bar>. \<exists>y w. x < y \<and> x < z \<and> x > w \<and> 3 * x < w + y"
   130   by ferrack
   131 
   132 lemma "\<forall>(x::'a::linordered_field) y. \<exists>z. \<forall>w. \<bar>x - y\<bar> = \<bar>z - w\<bar> \<longrightarrow> z < x \<and> w \<noteq> y"
   133   by ferrack
   134 
   135 lemma "\<exists>y. \<forall>x::'a::linordered_field. \<exists>z w.
   136   min (5 * x + 3 * z) (17 * w) + 5 * \<bar>y - 8 * x + z\<bar> \<le> max y (7 * z - x + w)"
   137   by ferrack
   138 
   139 lemma "\<exists>(x::'a::linordered_field) z. \<forall>w \<ge> 13 * x - 4 * z. \<exists>y. w \<ge> \<bar>x\<bar> + \<bar>y\<bar> + z"
   140   by ferrack
   141 
   142 lemma "\<exists>x::'a::linordered_field. \<forall>y < x. \<exists>z > x + y.
   143   \<forall>w. 5 * w + 10 * x - z \<ge> y \<longrightarrow> w + 7 * x + 3 * z \<ge> 2 * y"
   144   by ferrack
   145 
   146 lemma "\<exists>x::'a::linordered_field. \<forall>y. \<exists>z > y.
   147   \<forall>w. w < 13 \<longrightarrow> w + 10 * x - z \<ge> y \<longrightarrow> 5 * w + 7 * x + 13 * z \<ge> 2 * y"
   148   by ferrack
   149 
   150 lemma "\<exists>(x::'a::linordered_field) y z w.
   151   min (5 * x + 3 * z) (17 * w) + 5 * \<bar>y - 8 * x + z\<bar> \<le> max y (7 * z - x + w)"
   152   by ferrack
   153 
   154 lemma "\<forall>x::'a::linordered_field. \<exists>y. \<forall>z>19. y \<le> x + z \<and> (\<exists>w. \<bar>y - x\<bar> < w)"
   155   by ferrack
   156 
   157 lemma "\<forall>x::'a::linordered_field. \<exists>y. \<forall>z>19. y \<le> x + z \<and> (\<exists>w. \<bar>x + z\<bar> < w - y)"
   158   by ferrack
   159 
   160 lemma "\<forall>x::'a::linordered_field. \<exists>y.
   161   \<bar>y\<bar> \<noteq> \<bar>x\<bar> \<and> (\<forall>z > max x y. \<exists>w. w \<noteq> y \<and> w \<noteq> z \<and> 3 * w - z \<ge> x + y)"
   162   by ferrack
   163 
   164 end