src/HOL/Library/Sum_Of_Squares.thy
author huffman
Thu Jun 11 09:03:24 2009 -0700 (2009-06-11)
changeset 31563 ded2364d14d4
parent 31512 27118561c2e0
child 32268 d50f0cb67578
permissions -rw-r--r--
cleaned up some proofs
     1 (* Title:      Library/Sum_Of_Squares
     2    Author:     Amine Chaieb, University of Cambridge
     3 *)
     4 
     5 header {* A decision method for universal multivariate real arithmetic with addition, 
     6           multiplication and ordering using semidefinite programming*}
     7 theory Sum_Of_Squares
     8   imports Complex_Main (* "~~/src/HOL/Decision_Procs/Dense_Linear_Order" *)
     9   uses "positivstellensatz.ML" "sum_of_squares.ML"
    10   begin
    11 
    12 (* Note: 
    13 
    14 In order to use the method sos, install CSDP (https://projects.coin-or.org/Csdp/) and put the executable csdp on your path. 
    15 
    16 *)
    17 
    18 method_setup sos = {* Scan.succeed (SIMPLE_METHOD' o Sos.sos_tac) *} 
    19   "Prove universal problems over the reals using sums of squares"
    20 
    21 text{* Tests -- commented since they work only when csdp is installed -- see above *}
    22 (*
    23 lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x \<Longrightarrow> a < 0" by sos
    24 
    25 lemma "a1 >= 0 & a2 >= 0 \<and> (a1 * a1 + a2 * a2 = b1 * b1 + b2 * b2 + 2) \<and> (a1 * b1 + a2 * b2 = 0) --> a1 * a2 - b1 * b2 >= (0::real)" by sos
    26 
    27 lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x --> a < 0" by sos
    28 
    29 lemma "(0::real) <= x & x <= 1 & 0 <= y & y <= 1  --> x^2 + y^2 < 1 |(x - 1)^2 + y^2 < 1 | x^2 + (y - 1)^2 < 1 | (x - 1)^2 + (y - 1)^2 < 1" by sos
    30 
    31 lemma "(0::real) <= x & 0 <= y & 0 <= z & x + y + z <= 3 --> x * y + x * z + y * z >= 3 * x * y * z" by sos
    32 
    33 lemma "((x::real)^2 + y^2 + z^2 = 1) --> (x + y + z)^2 <= 3" by sos
    34 
    35 lemma "(w^2 + x^2 + y^2 + z^2 = 1) --> (w + x + y + z)^2 <= (4::real)" by sos
    36 
    37 lemma "(x::real) >= 1 & y >= 1 --> x * y >= x + y - 1" by sos
    38 
    39 lemma "(x::real) > 1 & y > 1 --> x * y > x + y - 1" by sos; 
    40 
    41 lemma "abs(x) <= 1 --> abs(64 * x^7 - 112 * x^5 + 56 * x^3 - 7 * x) <= (1::real)" by sos  
    42 *)
    43 (* ------------------------------------------------------------------------- *)
    44 (* One component of denominator in dodecahedral example.                     *)
    45 (* ------------------------------------------------------------------------- *)
    46 (*
    47 lemma "2 <= x & x <= 125841 / 50000 & 2 <= y & y <= 125841 / 50000 & 2 <= z & z <= 125841 / 50000 --> 2 * (x * z + x * y + y * z) - (x * x + y * y + z * z) >= (0::real)" by sos;
    48 *)
    49 (* ------------------------------------------------------------------------- *)
    50 (* Over a larger but simpler interval.                                       *)
    51 (* ------------------------------------------------------------------------- *)
    52 (*
    53 lemma "(2::real) <= x & x <= 4 & 2 <= y & y <= 4 & 2 <= z & z <= 4 --> 0 <= 2 * (x * z + x * y + y * z) - (x * x + y * y + z * z)" by sos
    54 *)
    55 (* ------------------------------------------------------------------------- *)
    56 (* We can do 12. I think 12 is a sharp bound; see PP's certificate.          *)
    57 (* ------------------------------------------------------------------------- *)
    58 (*
    59 lemma "2 <= (x::real) & x <= 4 & 2 <= y & y <= 4 & 2 <= z & z <= 4 --> 12 <= 2 * (x * z + x * y + y * z) - (x * x + y * y + z * z)" by sos
    60 *)
    61 
    62 (* ------------------------------------------------------------------------- *)
    63 (* Inequality from sci.math (see "Leon-Sotelo, por favor").                  *)
    64 (* ------------------------------------------------------------------------- *)
    65 (*
    66 lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x + y <= x^2 + y^2" by sos 
    67 
    68 lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x * y * (x + y) <= x^2 + y^2" by sos 
    69 
    70 lemma "0 <= (x::real) & 0 <= y --> x * y * (x + y)^2 <= (x^2 + y^2)^2" by sos
    71 
    72 lemma "(0::real) <= a & 0 <= b & 0 <= c & c * (2 * a + b)^3/ 27 <= x \<longrightarrow> c * a^2 * b <= x" by sos
    73  
    74 lemma "(0::real) < x --> 0 < 1 + x + x^2" by sos
    75 
    76 lemma "(0::real) <= x --> 0 < 1 + x + x^2" by sos
    77 
    78 lemma "(0::real) < 1 + x^2" by sos
    79 
    80 lemma "(0::real) <= 1 + 2 * x + x^2" by sos
    81 
    82 lemma "(0::real) < 1 + abs x" by sos
    83 
    84 lemma "(0::real) < 1 + (1 + x)^2 * (abs x)" by sos
    85 
    86 
    87 
    88 lemma "abs ((1::real) + x^2) = (1::real) + x^2" by sos
    89 lemma "(3::real) * x + 7 * a < 4 \<and> 3 < 2 * x \<longrightarrow> a < 0" by sos
    90 
    91 lemma "(0::real) < x --> 1 < y --> y * x <= z --> x < z" by sos
    92 lemma "(1::real) < x --> x^2 < y --> 1 < y" by sos
    93 lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)" by sos
    94 lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)" by sos
    95 lemma "((a::real) * x^2 + b * x + c = 0) --> b^2 >= 4 * a * c" by sos
    96 lemma "(0::real) <= b & 0 <= c & 0 <= x & 0 <= y & (x^2 = c) & (y^2 = a^2 * c + b) --> a * c <= y * x" by sos
    97 lemma "abs(x - z) <= e & abs(y - z) <= e & 0 <= u & 0 <= v & (u + v = 1) --> abs((u * x + v * y) - z) <= (e::real)" by sos
    98 *)
    99 (*
   100 lemma "((x::real) - y - 2 * x^4 = 0) & 0 <= x & x <= 2 & 0 <= y & y <= 3 --> y^2 - 7 * y - 12 * x + 17 >= 0" by sos *) (* Too hard?*)
   101 (*
   102 lemma "(0::real) <= x --> (1 + x + x^2)/(1 + x^2) <= 1 + x"
   103 apply sos
   104 done
   105 
   106 lemma "(0::real) <= x --> 1 - x <= 1 / (1 + x + x^2)"
   107 apply sos
   108 done
   109 
   110 lemma "(x::real) <= 1 / 2 --> - x - 2 * x^2 <= - x / (1 - x)"
   111 apply sos
   112 done 
   113 
   114 lemma "4*r^2 = p^2 - 4*q & r >= (0::real) & x^2 + p*x + q = 0 --> 2*(x::real) = - p + 2*r | 2*x = -p - 2*r" by sos
   115 *)
   116 
   117 end