--- a/src/HOL/Library/Sum_of_Squares_Remote.thy Mon Aug 27 14:34:54 2012 +0200
+++ b/src/HOL/Library/Sum_of_Squares_Remote.thy Mon Aug 27 16:00:42 2012 +0200
@@ -33,102 +33,4 @@
lemma "(x::real) >= 1 & y >= 1 --> x * y >= x + y - 1"
by (sos remote_csdp)
-lemma "(x::real) > 1 & y > 1 --> x * y > x + y - 1"
- by (sos remote_csdp)
-
-lemma "abs(x) <= 1 --> abs(64 * x^7 - 112 * x^5 + 56 * x^3 - 7 * x) <= (1::real)"
- by (sos remote_csdp)
-
-(* ------------------------------------------------------------------------- *)
-(* One component of denominator in dodecahedral example. *)
-(* ------------------------------------------------------------------------- *)
-
-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 remote_csdp)
-
-(* ------------------------------------------------------------------------- *)
-(* Over a larger but simpler interval. *)
-(* ------------------------------------------------------------------------- *)
-
-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 remote_csdp)
-
-(* ------------------------------------------------------------------------- *)
-(* We can do 12. I think 12 is a sharp bound; see PP's certificate. *)
-(* ------------------------------------------------------------------------- *)
-
-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 remote_csdp)
-
-(* ------------------------------------------------------------------------- *)
-(* Inequality from sci.math (see "Leon-Sotelo, por favor"). *)
-(* ------------------------------------------------------------------------- *)
-
-lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x + y <= x^2 + y^2"
- by (sos remote_csdp)
-
-lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x * y * (x + y) <= x^2 + y^2"
- by (sos remote_csdp)
-
-lemma "0 <= (x::real) & 0 <= y --> x * y * (x + y)^2 <= (x^2 + y^2)^2"
- by (sos remote_csdp)
-
-lemma "(0::real) <= a & 0 <= b & 0 <= c & c * (2 * a + b)^3/ 27 <= x \<longrightarrow> c * a^2 * b <= x"
- by (sos remote_csdp)
-
-lemma "(0::real) < x --> 0 < 1 + x + x^2"
- by (sos remote_csdp)
-
-lemma "(0::real) <= x --> 0 < 1 + x + x^2"
- by (sos remote_csdp)
-
-lemma "(0::real) < 1 + x^2"
- by (sos remote_csdp)
-
-lemma "(0::real) <= 1 + 2 * x + x^2"
- by (sos remote_csdp)
-
-lemma "(0::real) < 1 + abs x"
- by (sos remote_csdp)
-
-lemma "(0::real) < 1 + (1 + x)^2 * (abs x)"
- by (sos remote_csdp)
-
-
-
-lemma "abs ((1::real) + x^2) = (1::real) + x^2"
- by (sos remote_csdp)
-lemma "(3::real) * x + 7 * a < 4 \<and> 3 < 2 * x \<longrightarrow> a < 0"
- by (sos remote_csdp)
-
-lemma "(0::real) < x --> 1 < y --> y * x <= z --> x < z"
- by (sos remote_csdp)
-lemma "(1::real) < x --> x^2 < y --> 1 < y"
- by (sos remote_csdp)
-lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)"
- by (sos remote_csdp)
-lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)"
- by (sos remote_csdp)
-lemma "((a::real) * x^2 + b * x + c = 0) --> b^2 >= 4 * a * c"
- by (sos remote_csdp)
-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 remote_csdp)
-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 remote_csdp)
-
-
-(* 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?*)
-
-lemma "(0::real) <= x --> (1 + x + x^2)/(1 + x^2) <= 1 + x"
- by (sos remote_csdp)
-
-lemma "(0::real) <= x --> 1 - x <= 1 / (1 + x + x^2)"
- by (sos remote_csdp)
-
-lemma "(x::real) <= 1 / 2 --> - x - 2 * x^2 <= - x / (1 - x)"
- by (sos remote_csdp)
-
-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 remote_csdp)
-
end