src/HOL/Library/Sum_of_Squares_Remote.thy
 changeset 48934 f9a800f21434 parent 48932 c6e679443adc child 58881 b9556a055632
```     1.1 --- a/src/HOL/Library/Sum_of_Squares_Remote.thy	Mon Aug 27 14:34:54 2012 +0200
1.2 +++ b/src/HOL/Library/Sum_of_Squares_Remote.thy	Mon Aug 27 16:00:42 2012 +0200
1.3 @@ -33,102 +33,4 @@
1.4  lemma "(x::real) >= 1 & y >= 1 --> x * y >= x + y - 1"
1.5    by (sos remote_csdp)
1.6
1.7 -lemma "(x::real) > 1 & y > 1 --> x * y > x + y - 1"
1.8 -  by (sos remote_csdp)
1.9 -
1.10 -lemma "abs(x) <= 1 --> abs(64 * x^7 - 112 * x^5 + 56 * x^3 - 7 * x) <= (1::real)"
1.11 -  by (sos remote_csdp)
1.12 -
1.13 -(* ------------------------------------------------------------------------- *)
1.14 -(* One component of denominator in dodecahedral example.                     *)
1.15 -(* ------------------------------------------------------------------------- *)
1.16 -
1.17 -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)"
1.18 -  by (sos remote_csdp)
1.19 -
1.20 -(* ------------------------------------------------------------------------- *)
1.21 -(* Over a larger but simpler interval.                                       *)
1.22 -(* ------------------------------------------------------------------------- *)
1.23 -
1.24 -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)"
1.25 -  by (sos remote_csdp)
1.26 -
1.27 -(* ------------------------------------------------------------------------- *)
1.28 -(* We can do 12. I think 12 is a sharp bound; see PP's certificate.          *)
1.29 -(* ------------------------------------------------------------------------- *)
1.30 -
1.31 -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)"
1.32 -  by (sos remote_csdp)
1.33 -
1.34 -(* ------------------------------------------------------------------------- *)
1.35 -(* Inequality from sci.math (see "Leon-Sotelo, por favor").                  *)
1.36 -(* ------------------------------------------------------------------------- *)
1.37 -
1.38 -lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x + y <= x^2 + y^2"
1.39 -  by (sos remote_csdp)
1.40 -
1.41 -lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x * y * (x + y) <= x^2 + y^2"
1.42 -  by (sos remote_csdp)
1.43 -
1.44 -lemma "0 <= (x::real) & 0 <= y --> x * y * (x + y)^2 <= (x^2 + y^2)^2"
1.45 -  by (sos remote_csdp)
1.46 -
1.47 -lemma "(0::real) <= a & 0 <= b & 0 <= c & c * (2 * a + b)^3/ 27 <= x \<longrightarrow> c * a^2 * b <= x"
1.48 -  by (sos remote_csdp)
1.49 -
1.50 -lemma "(0::real) < x --> 0 < 1 + x + x^2"
1.51 -  by (sos remote_csdp)
1.52 -
1.53 -lemma "(0::real) <= x --> 0 < 1 + x + x^2"
1.54 -  by (sos remote_csdp)
1.55 -
1.56 -lemma "(0::real) < 1 + x^2"
1.57 -  by (sos remote_csdp)
1.58 -
1.59 -lemma "(0::real) <= 1 + 2 * x + x^2"
1.60 -  by (sos remote_csdp)
1.61 -
1.62 -lemma "(0::real) < 1 + abs x"
1.63 -  by (sos remote_csdp)
1.64 -
1.65 -lemma "(0::real) < 1 + (1 + x)^2 * (abs x)"
1.66 -  by (sos remote_csdp)
1.67 -
1.68 -
1.69 -
1.70 -lemma "abs ((1::real) + x^2) = (1::real) + x^2"
1.71 -  by (sos remote_csdp)
1.72 -lemma "(3::real) * x + 7 * a < 4 \<and> 3 < 2 * x \<longrightarrow> a < 0"
1.73 -  by (sos remote_csdp)
1.74 -
1.75 -lemma "(0::real) < x --> 1 < y --> y * x <= z --> x < z"
1.76 -  by (sos remote_csdp)
1.77 -lemma "(1::real) < x --> x^2 < y --> 1 < y"
1.78 -  by (sos remote_csdp)
1.79 -lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)"
1.80 -  by (sos remote_csdp)
1.81 -lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)"
1.82 -  by (sos remote_csdp)
1.83 -lemma "((a::real) * x^2 + b * x + c = 0) --> b^2 >= 4 * a * c"
1.84 -  by (sos remote_csdp)
1.85 -lemma "(0::real) <= b & 0 <= c & 0 <= x & 0 <= y & (x^2 = c) & (y^2 = a^2 * c + b) --> a * c <= y * x"
1.86 -  by (sos remote_csdp)
1.87 -lemma "abs(x - z) <= e & abs(y - z) <= e & 0 <= u & 0 <= v & (u + v = 1) --> abs((u * x + v * y) - z) <= (e::real)"
1.88 -  by (sos remote_csdp)
1.89 -
1.90 -
1.91 -(* 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?*)
1.92 -
1.93 -lemma "(0::real) <= x --> (1 + x + x^2)/(1 + x^2) <= 1 + x"
1.94 -  by (sos remote_csdp)
1.95 -
1.96 -lemma "(0::real) <= x --> 1 - x <= 1 / (1 + x + x^2)"
1.97 -  by (sos remote_csdp)
1.98 -
1.99 -lemma "(x::real) <= 1 / 2 --> - x - 2 * x^2 <= - x / (1 - x)"
1.100 -  by (sos remote_csdp)
1.101 -
1.102 -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"
1.103 -  by (sos remote_csdp)
1.104 -
1.105  end
```