src/HOL/ex/PresburgerEx.thy
author obua
Mon Apr 10 16:00:34 2006 +0200 (2006-04-10)
changeset 19404 9bf2cdc9e8e8
parent 17388 495c799df31d
child 19824 fafceecebef0
permissions -rw-r--r--
Moved stuff from Ring_and_Field to Matrix
     1 (*  Title:      HOL/ex/PresburgerEx.thy
     2     ID:         $Id$
     3     Author:     Amine Chaieb, TU Muenchen
     4 *)
     5 
     6 header {* Some examples for Presburger Arithmetic *}
     7 
     8 theory PresburgerEx imports Main begin
     9 
    10 theorem "(\<forall>(y::int). 3 dvd y) ==> \<forall>(x::int). b < x --> a \<le> x"
    11   by presburger
    12 
    13 theorem "!! (y::int) (z::int) (n::int). 3 dvd z ==> 2 dvd (y::int) ==>
    14   (\<exists>(x::int).  2*x =  y) & (\<exists>(k::int). 3*k = z)"
    15   by presburger
    16 
    17 theorem "!! (y::int) (z::int) n. Suc(n::nat) < 6 ==>  3 dvd z ==>
    18   2 dvd (y::int) ==> (\<exists>(x::int).  2*x =  y) & (\<exists>(k::int). 3*k = z)"
    19   by presburger
    20 
    21 theorem "\<forall>(x::nat). \<exists>(y::nat). (0::nat) \<le> 5 --> y = 5 + x "
    22   by presburger
    23 
    24 text{*Very slow: about 55 seconds on a 1.8GHz machine.*}
    25 theorem "\<forall>(x::nat). \<exists>(y::nat). y = 5 + x | x div 6 + 1= 2"
    26   by presburger
    27 
    28 theorem "\<exists>(x::int). 0 < x"
    29   by presburger
    30 
    31 theorem "\<forall>(x::int) y. x < y --> 2 * x + 1 < 2 * y"
    32   by presburger
    33  
    34 theorem "\<forall>(x::int) y. 2 * x + 1 \<noteq> 2 * y"
    35   by presburger
    36  
    37 theorem "\<exists>(x::int) y. 0 < x  & 0 \<le> y  & 3 * x - 5 * y = 1"
    38   by presburger
    39 
    40 theorem "~ (\<exists>(x::int) (y::int) (z::int). 4*x + (-6::int)*y = 1)"
    41   by presburger
    42 
    43 theorem "\<forall>(x::int). b < x --> a \<le> x"
    44   apply (presburger (no_quantify))
    45   oops
    46 
    47 theorem "~ (\<exists>(x::int). False)"
    48   by presburger
    49 
    50 theorem "\<forall>(x::int). (a::int) < 3 * x --> b < 3 * x"
    51   apply (presburger (no_quantify))
    52   oops
    53 
    54 theorem "\<forall>(x::int). (2 dvd x) --> (\<exists>(y::int). x = 2*y)"
    55   by presburger 
    56 
    57 theorem "\<forall>(x::int). (2 dvd x) --> (\<exists>(y::int). x = 2*y)"
    58   by presburger 
    59 
    60 theorem "\<forall>(x::int). (2 dvd x) = (\<exists>(y::int). x = 2*y)"
    61   by presburger 
    62 
    63 theorem "\<forall>(x::int). ((2 dvd x) = (\<forall>(y::int). x \<noteq> 2*y + 1))"
    64   by presburger 
    65 
    66 theorem "~ (\<forall>(x::int). 
    67             ((2 dvd x) = (\<forall>(y::int). x \<noteq> 2*y+1) | 
    68              (\<exists>(q::int) (u::int) i. 3*i + 2*q - u < 17)
    69              --> 0 < x | ((~ 3 dvd x) &(x + 8 = 0))))"
    70   by presburger
    71  
    72 theorem "~ (\<forall>(i::int). 4 \<le> i --> (\<exists>x y. 0 \<le> x & 0 \<le> y & 3 * x + 5 * y = i))"
    73   by presburger
    74 
    75 theorem "\<forall>(i::int). 8 \<le> i --> (\<exists>x y. 0 \<le> x & 0 \<le> y & 3 * x + 5 * y = i)"
    76   by presburger
    77 
    78 theorem "\<exists>(j::int). \<forall>i. j \<le> i --> (\<exists>x y. 0 \<le> x & 0 \<le> y & 3 * x + 5 * y = i)"
    79   by presburger
    80 
    81 theorem "~ (\<forall>j (i::int). j \<le> i --> (\<exists>x y. 0 \<le> x & 0 \<le> y & 3 * x + 5 * y = i))"
    82   by presburger
    83 
    84 text{*Very slow: about 80 seconds on a 1.8GHz machine.*}
    85 theorem "(\<exists>m::nat. n = 2 * m) --> (n + 1) div 2 = n div 2"
    86   by presburger
    87 
    88 theorem "(\<exists>m::int. n = 2 * m) --> (n + 1) div 2 = n div 2"
    89   by presburger
    90 
    91 end