src/HOL/ex/Dense_Linear_Order_Ex.thy
 author nipkow Fri Mar 06 17:38:47 2009 +0100 (2009-03-06) changeset 30313 b2441b0c8d38 parent 29823 0ab754d13ccd permissions -rw-r--r--
```     1 (* Author:     Amine Chaieb, TU Muenchen *)
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```     2
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```     3 header {* Examples for Ferrante and Rackoff's quantifier elimination procedure *}
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```     4
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```     5 theory Dense_Linear_Order_Ex
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```     6 imports "~~/src/HOL/Decision_Procs/Dense_Linear_Order" Main
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```     7 begin
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```     8
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```     9 lemma
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```    10   "\<exists>(y::'a::{ordered_field,recpower,number_ring, division_by_zero}) <2. x + 3* y < 0 \<and> x - y >0"
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```    11   by ferrack
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```    12
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```    13 lemma "~ (ALL x (y::'a::{ordered_field,recpower,number_ring, division_by_zero}). x < y --> 10*x < 11*y)"
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```    14   by ferrack
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```    15
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```    16 lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. x < y --> (10*(x + 5*y + -1) < 60*y)"
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```    17   by ferrack
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```    18
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```    19 lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. x ~= y --> x < y"
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```    20   by ferrack
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```    21
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```    22 lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. (x ~= y & 10*x ~= 9*y & 10*x < y) --> x < y"
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```    23   by ferrack
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```    24
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```    25 lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. (x ~= y & 5*x <= y) --> 500*x <= 100*y"
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```    26   by ferrack
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```    27
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```    28 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)"
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```    29   by ferrack
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```    30
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```    31 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)"
```
```    32   by ferrack
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```    33
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```    34 lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}). (0 < x & x < 1) --> (ALL y > 1. x + y ~= 1)"
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```    35   by ferrack
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```    36
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```    37 lemma "EX x. (ALL (y::'a::{ordered_field,recpower,number_ring, division_by_zero}). y < 2 -->  2*(y - x) \<le> 0 )"
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```    38   by ferrack
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```    39
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```    40 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)"
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```    41   by ferrack
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```    42
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```    43 lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y z. x + y < z --> y >= z --> x < 0"
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```    44   by ferrack
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```    45
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```    46 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"
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```    47   by ferrack
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```    48
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```    49 lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y z. abs (x + y) <= z --> (abs z = z)"
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```    50   by ferrack
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```    51
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```    52 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"
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```    53   by ferrack
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```    54
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```    55 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))"
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```    56   by ferrack
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```    57
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```    58 lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. x < y --> (EX z>0. x+z = y)"
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```    59   by ferrack
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```    60
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```    61 lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. x < y --> (EX z>0. x+z = y)"
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```    62   by ferrack
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```    63
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```    64 lemma "ALL (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. (EX z>0. abs (x - y) <= z )"
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```    65   by ferrack
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```    66
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```    67 lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. (ALL z<0. (z < x --> z <= y) & (z > y --> z >= x))"
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```    68   by ferrack
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```    69
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```    70 lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. (ALL z>=0. abs (3*x+7*y) <= 2*z + 1)"
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```    71   by ferrack
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```    72
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```    73 lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}) y. (ALL z<0. (z < x --> z <= y) & (z > y --> z >= x))"
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```    74   by ferrack
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```    75
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```    76 lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero})>0. (ALL y. (EX z. 13* abs z \<noteq> abs (12*y - x) & 5*x - 3*(abs y) <= 7*z))"
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```    77   by ferrack
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```    78
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```    79 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) \<noteq> 0 \<longrightarrow> (EX z. 5*x - 3*abs y <= 7*z))"
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```    80   by ferrack
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```    81
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```    82 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))"
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```    83   by ferrack
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```    84
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```    85 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) ))"
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```    86   by ferrack
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```    87
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```    88 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))"
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```    89   by ferrack
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```    90
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```    91 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"
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```    92   by ferrack
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```    93
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```    94 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"
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```    95   by ferrack
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```    96
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```    97 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)"
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```    98   by ferrack
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```    99
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```   100 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)"
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```   101   by ferrack
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```   102
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```   103 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)"
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```   104   by ferrack
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```   105
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```   106 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))"
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```   107   by ferrack
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```   108
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```   109 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)"
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```   110   by ferrack
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```   111
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```   112 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))"
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```   113   by ferrack
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```   114
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```   115 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)"
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```   116   by ferrack
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```   117
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```   118 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))"
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```   119   by ferrack
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```   120
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```   121 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))"
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```   122   by ferrack
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```   123
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```   124 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))"
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```   125   by ferrack
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```   126
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```   127 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)))"
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```   128   by ferrack
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```   129
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```   130 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))"
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```   131   by ferrack
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```   132
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```   133 lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}). (ALL y < x. (EX z > (x+y).
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```   134   (ALL w. 5*w + 10*x - z >= y --> w + 7*x + 3*z >= 2*y)))"
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```   135   by ferrack
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```   136
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```   137 lemma "EX (x::'a::{ordered_field,recpower,number_ring, division_by_zero}). (ALL y. (EX z > y.
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```   138   (ALL w . w < 13 --> w + 10*x - z >= y --> 5*w + 7*x + 13*z >= 2*y)))"
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```   139   by ferrack
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```   140
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```   141 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)"
```
```   142   by ferrack
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```   143
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```   144 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)))"
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```   145   by ferrack
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```   146
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```   147 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)))"
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```   148   by ferrack
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```   149
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```   150 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)))"
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```   151   by ferrack
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```   152
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```   153 end
```