src/HOL/Library/Sum_of_Squares_Remote.thy
changeset 48932 c6e679443adc
child 48934 f9a800f21434
     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/src/HOL/Library/Sum_of_Squares_Remote.thy	Sun Aug 19 17:45:07 2012 +0200
     1.3 @@ -0,0 +1,134 @@
     1.4 +(*  Title:      HOL/Library/Sum_of_Squares_Remote.thy
     1.5 +    Author:     Amine Chaieb, University of Cambridge
     1.6 +    Author:     Philipp Meyer, TU Muenchen
     1.7 +*)
     1.8 +
     1.9 +header {* Examples with remote CSDP *}
    1.10 +
    1.11 +theory Sum_of_Squares_Remote
    1.12 +imports Sum_of_Squares
    1.13 +begin
    1.14 +
    1.15 +lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x \<Longrightarrow> a < 0"
    1.16 +  by (sos remote_csdp)
    1.17 +
    1.18 +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)"
    1.19 +  by (sos remote_csdp)
    1.20 +
    1.21 +lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x --> a < 0"
    1.22 +  by (sos remote_csdp)
    1.23 +
    1.24 +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"
    1.25 +  by (sos remote_csdp)
    1.26 +
    1.27 +lemma "(0::real) <= x & 0 <= y & 0 <= z & x + y + z <= 3 --> x * y + x * z + y * z >= 3 * x * y * z"
    1.28 +  by (sos remote_csdp)
    1.29 +
    1.30 +lemma "((x::real)^2 + y^2 + z^2 = 1) --> (x + y + z)^2 <= 3"
    1.31 +  by (sos remote_csdp)
    1.32 +
    1.33 +lemma "(w^2 + x^2 + y^2 + z^2 = 1) --> (w + x + y + z)^2 <= (4::real)"
    1.34 +  by (sos remote_csdp)
    1.35 +
    1.36 +lemma "(x::real) >= 1 & y >= 1 --> x * y >= x + y - 1"
    1.37 +  by (sos remote_csdp)
    1.38 +
    1.39 +lemma "(x::real) > 1 & y > 1 --> x * y > x + y - 1"
    1.40 +  by (sos remote_csdp)
    1.41 +
    1.42 +lemma "abs(x) <= 1 --> abs(64 * x^7 - 112 * x^5 + 56 * x^3 - 7 * x) <= (1::real)"
    1.43 +  by (sos remote_csdp)
    1.44 +
    1.45 +(* ------------------------------------------------------------------------- *)
    1.46 +(* One component of denominator in dodecahedral example.                     *)
    1.47 +(* ------------------------------------------------------------------------- *)
    1.48 +
    1.49 +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)"
    1.50 +  by (sos remote_csdp)
    1.51 +
    1.52 +(* ------------------------------------------------------------------------- *)
    1.53 +(* Over a larger but simpler interval.                                       *)
    1.54 +(* ------------------------------------------------------------------------- *)
    1.55 +
    1.56 +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)"
    1.57 +  by (sos remote_csdp)
    1.58 +
    1.59 +(* ------------------------------------------------------------------------- *)
    1.60 +(* We can do 12. I think 12 is a sharp bound; see PP's certificate.          *)
    1.61 +(* ------------------------------------------------------------------------- *)
    1.62 +
    1.63 +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)"
    1.64 +  by (sos remote_csdp)
    1.65 +
    1.66 +(* ------------------------------------------------------------------------- *)
    1.67 +(* Inequality from sci.math (see "Leon-Sotelo, por favor").                  *)
    1.68 +(* ------------------------------------------------------------------------- *)
    1.69 +
    1.70 +lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x + y <= x^2 + y^2"
    1.71 +  by (sos remote_csdp)
    1.72 +
    1.73 +lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x * y * (x + y) <= x^2 + y^2"
    1.74 +  by (sos remote_csdp)
    1.75 +
    1.76 +lemma "0 <= (x::real) & 0 <= y --> x * y * (x + y)^2 <= (x^2 + y^2)^2"
    1.77 +  by (sos remote_csdp)
    1.78 +
    1.79 +lemma "(0::real) <= a & 0 <= b & 0 <= c & c * (2 * a + b)^3/ 27 <= x \<longrightarrow> c * a^2 * b <= x"
    1.80 +  by (sos remote_csdp)
    1.81 +
    1.82 +lemma "(0::real) < x --> 0 < 1 + x + x^2"
    1.83 +  by (sos remote_csdp)
    1.84 +
    1.85 +lemma "(0::real) <= x --> 0 < 1 + x + x^2"
    1.86 +  by (sos remote_csdp)
    1.87 +
    1.88 +lemma "(0::real) < 1 + x^2"
    1.89 +  by (sos remote_csdp)
    1.90 +
    1.91 +lemma "(0::real) <= 1 + 2 * x + x^2"
    1.92 +  by (sos remote_csdp)
    1.93 +
    1.94 +lemma "(0::real) < 1 + abs x"
    1.95 +  by (sos remote_csdp)
    1.96 +
    1.97 +lemma "(0::real) < 1 + (1 + x)^2 * (abs x)"
    1.98 +  by (sos remote_csdp)
    1.99 +
   1.100 +
   1.101 +
   1.102 +lemma "abs ((1::real) + x^2) = (1::real) + x^2"
   1.103 +  by (sos remote_csdp)
   1.104 +lemma "(3::real) * x + 7 * a < 4 \<and> 3 < 2 * x \<longrightarrow> a < 0"
   1.105 +  by (sos remote_csdp)
   1.106 +
   1.107 +lemma "(0::real) < x --> 1 < y --> y * x <= z --> x < z"
   1.108 +  by (sos remote_csdp)
   1.109 +lemma "(1::real) < x --> x^2 < y --> 1 < y"
   1.110 +  by (sos remote_csdp)
   1.111 +lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)"
   1.112 +  by (sos remote_csdp)
   1.113 +lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)"
   1.114 +  by (sos remote_csdp)
   1.115 +lemma "((a::real) * x^2 + b * x + c = 0) --> b^2 >= 4 * a * c"
   1.116 +  by (sos remote_csdp)
   1.117 +lemma "(0::real) <= b & 0 <= c & 0 <= x & 0 <= y & (x^2 = c) & (y^2 = a^2 * c + b) --> a * c <= y * x"
   1.118 +  by (sos remote_csdp)
   1.119 +lemma "abs(x - z) <= e & abs(y - z) <= e & 0 <= u & 0 <= v & (u + v = 1) --> abs((u * x + v * y) - z) <= (e::real)"
   1.120 +  by (sos remote_csdp)
   1.121 +
   1.122 +
   1.123 +(* 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?*)
   1.124 +
   1.125 +lemma "(0::real) <= x --> (1 + x + x^2)/(1 + x^2) <= 1 + x"
   1.126 +  by (sos remote_csdp)
   1.127 +
   1.128 +lemma "(0::real) <= x --> 1 - x <= 1 / (1 + x + x^2)"
   1.129 +  by (sos remote_csdp)
   1.130 +
   1.131 +lemma "(x::real) <= 1 / 2 --> - x - 2 * x^2 <= - x / (1 - x)"
   1.132 +  by (sos remote_csdp)
   1.133 +
   1.134 +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"
   1.135 +  by (sos remote_csdp)
   1.136 +
   1.137 +end