src/HOL/Library/Sum_Of_Squares.thy
 changeset 31119 2532bb2d65c7 child 31131 d9752181691a
```     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/Library/Sum_Of_Squares.thy	Tue May 12 17:32:49 2009 +0100
1.3 @@ -0,0 +1,114 @@
1.4 +(* Title:      Library/Sum_Of_Squares
1.5 +   Author:     Amine Chaieb, University of Cambridge
1.6 +*)
1.7 +
1.8 +header {* A decision method for universal multivariate real arithmetic with addition,
1.9 +          multiplication and ordering using semidefinite programming*}
1.10 +theory Sum_Of_Squares
1.11 +  imports Complex_Main (* "~~/src/HOL/Decision_Procs/Dense_Linear_Order" *)
1.12 +  uses "positivstellensatz.ML" "sum_of_squares.ML"
1.13 +  begin
1.14 +
1.15 +method_setup sos = {*
1.16 +let
1.17 + fun strip_all ct =
1.18 +  case term_of ct of
1.19 +   Const("all",_) \$ Abs (xn,xT,p) =>
1.20 +    let val (a,(v,t')) = (apsnd (Thm.dest_abs (SOME xn)) o Thm.dest_comb) ct
1.21 +    in apfst (cons v) (strip_all t')
1.22 +    end
1.23 + | _ => ([],ct)
1.24 +
1.25 + fun core_sos_conv ctxt t = Drule.arg_cong_rule @{cterm Trueprop} (Sos.real_sos ctxt (Thm.dest_arg t) RS @{thm Eq_TrueI})
1.26 + fun core_sos_tac ctxt = CSUBGOAL (fn (ct, i) =>
1.27 +   let val (avs, p) = strip_all ct
1.28 +       val th = standard (fold_rev forall_intr avs (Sos.real_sos ctxt (Thm.dest_arg p)))
1.29 +   in rtac th i end) (* CONVERSION o core_sos_conv *)
1.30 +in Scan.succeed (SIMPLE_METHOD' o core_sos_tac)
1.31 +end
1.32 +
1.33 +*} "Prove universal problems over the reals using sums of squares"
1.34 +
1.35 +text{* Tests -- commented since they work only when csdp is installed *}
1.36 +
1.37 +(*
1.38 +lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x --> a < 0" by sos
1.39 +
1.40 +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
1.41 +
1.42 +lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x --> a < 0" by sos
1.43 +
1.44 +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
1.45 +
1.46 +lemma "(0::real) <= x & 0 <= y & 0 <= z & x + y + z <= 3 --> x * y + x * z + y * z >= 3 * x * y * z" by sos
1.47 +
1.48 +lemma "((x::real)^2 + y^2 + z^2 = 1) --> (x + y + z)^2 <= 3" by sos
1.49 +
1.50 +lemma "(w^2 + x^2 + y^2 + z^2 = 1) --> (w + x + y + z)^2 <= (4::real)" by sos
1.51 +
1.52 +lemma "(x::real) >= 1 & y >= 1 --> x * y >= x + y - 1" by sos
1.53 +
1.54 +lemma "(x::real) > 1 & y > 1 --> x * y > x + y - 1" by sos;
1.55 +
1.56 +lemma "abs(x) <= 1 --> abs(64 * x^7 - 112 * x^5 + 56 * x^3 - 7 * x) <= (1::real)" by sos
1.57 +*)
1.58 +(* ------------------------------------------------------------------------- *)
1.59 +(* One component of denominator in dodecahedral example.                     *)
1.60 +(* ------------------------------------------------------------------------- *)
1.61 +(*
1.62 +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;
1.63 +*)
1.64 +(* ------------------------------------------------------------------------- *)
1.65 +(* Over a larger but simpler interval.                                       *)
1.66 +(* ------------------------------------------------------------------------- *)
1.67 +(*
1.68 +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
1.69 +*)
1.70 +(* ------------------------------------------------------------------------- *)
1.71 +(* We can do 12. I think 12 is a sharp bound; see PP's certificate.          *)
1.72 +(* ------------------------------------------------------------------------- *)
1.73 +(*
1.74 +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
1.75 +*)
1.76 +
1.77 +(* ------------------------------------------------------------------------- *)
1.78 +(* Inequality from sci.math (see "Leon-Sotelo, por favor").                  *)
1.79 +(* ------------------------------------------------------------------------- *)
1.80 +(*
1.81 +lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x + y <= x^2 + y^2" by sos
1.82 +
1.83 +lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x * y * (x + y) <= x^2 + y^2" by sos
1.84 +
1.85 +lemma "0 <= (x::real) & 0 <= y --> x * y * (x + y)^2 <= (x^2 + y^2)^2" by sos
1.86 +
1.87 +lemma "(0::real) <= a & 0 <= b & 0 <= c & c * (2 * a + b)^3/ 27 <= x \<longrightarrow> c * a^2 * b <= x" by sos
1.88 +
1.89 +lemma "(0::real) < x --> 0 < 1 + x + x^2" by sos
1.90 +
1.91 +lemma "(0::real) <= x --> 0 < 1 + x + x^2" by sos
1.92 +
1.93 +lemma "(0::real) < 1 + x^2" by sos
1.94 +
1.95 +lemma "(0::real) <= 1 + 2 * x + x^2" by sos
1.96 +
1.97 +lemma "(0::real) < 1 + abs x" by sos
1.98 +
1.99 +lemma "(0::real) < 1 + (1 + x)^2 * (abs x)" by sos
1.100 +
1.101 +
1.102 +
1.103 +lemma "abs ((1::real) + x^2) = (1::real) + x^2" by sos
1.104 +lemma "(3::real) * x + 7 * a < 4 \<and> 3 < 2 * x \<longrightarrow> a < 0" by sos
1.105 +
1.106 +lemma "(0::real) < x --> 1 < y --> y * x <= z --> x < z" by sos
1.107 +lemma "(1::real) < x --> x^2 < y --> 1 < y" by sos
1.108 +lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)" by sos
1.109 +lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)" by sos
1.110 +lemma "((a::real) * x^2 + b * x + c = 0) --> b^2 >= 4 * a * c" by sos
1.111 +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
1.112 +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
1.113 +*)
1.114 +(*
1.115 +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?*)
1.116 +
1.117 +end
```