Impressum SOS_Cert.thy   Sprache: Isabelle

(*  Title:      HOL/ex/SOS_Cert.thy
    Author:     Amine Chaieb, University of Cambridge
    Author:     Philipp Meyer, TU Muenchen

Examples for Sum_of_Squares: replay of certificates.
*)

theory SOS_Cert
imports "HOL-Library.Sum_of_Squares"
begin

lemma "(3::real) * x + 7 * a < 4 \ 3 < 2 * x \ a < 0"
  by (sos "((R<1 + (((A<1 * R<1) * (R<2 * [1]^2)) + (((A<0 * R<1) * (R<3 * [1]^2)) + ((A<=0 * R<1) * (R<14 * [1]^2))))))")

lemma "a1 \ 0 \ a2 \ 0 \ (a1 * a1 + a2 * a2 = b1 * b1 + b2 * b2 + 2) \ (a1 * b1 + a2 * b2 = 0) \
    a1 * a2 - b1 * b2 ≥ (0::real)"
  by (sos "(((A<0 * R<1) + (([~1/2*a1*b2 + ~1/2*a2*b1] * A=0) + (([~1/2*a1*a2 + 1/2*b1*b2] * A=1) + (((A<0 * R<1) * ((R<1/2 * [b2]^2) + (R<1/2 * [b1]^2))) + ((A<=0 * (A<=1 * R<1)) * ((R<1/2 * [b2]^2) + ((R<1/2 * [b1]^2) + ((R<1/2 * [a2]^2) + (R<1/2 * [a1]^2))))))))))")

lemma "(3::real) * x + 7 * a < 4 \ 3 < 2 * x \ a < 0"
  by (sos "((R<1 + (((A<1 * R<1) * (R<2 * [1]^2)) + (((A<0 * R<1) * (R<3 * [1]^2)) + ((A<=0 * R<1) * (R<14 * [1]^2))))))")

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 "((R<1 + (((A<=3 * (A<=4 * R<1)) * (R<1 * [1]^2)) + (((A<=2 * (A<=7 * R<1)) * (R<1 * [1]^2)) + (((A<=1 * (A<=6 * R<1)) * (R<1 * [1]^2)) + ((A<=0 * (A<=5 * R<1)) * (R<1 * [1]^2)))))))")

lemma "(0::real) \ x \ 0 \ y \ 0 \ z \ x + y + z \ 3 \ x * y + x * z + y * z \ 3 * x * y * z"
  by (sos "(((A<0 * R<1) + (((A<0 * R<1) * (R<1/2 * [1]^2)) + (((A<=2 * R<1) * (R<1/2 * [~1*x + y]^2)) + (((A<=1 * R<1) * (R<1/2 * [~1*x + z]^2)) + (((A<=1 * (A<=2 * (A<=3 * R<1))) * (R<1/2 * [1]^2)) + (((A<=0 * R<1) * (R<1/2 * [~1*y + z]^2)) + (((A<=0 * (A<=2 * (A<=3 * R<1))) * (R<1/2 * [1]^2)) + ((A<=0 * (A<=1 * (A<=3 * R<1))) * (R<1/2 * [1]^2))))))))))")

lemma "(x::real)\<^sup>2 + y\<^sup>2 + z\<^sup>2 = 1 \ (x + y + z)\<^sup>2 \ 3"
  by (sos "(((A<0 * R<1) + (([~3] * A=0) + (R<1 * ((R<2 * [~1/2*x + ~1/2*y + z]^2) + (R<3/2 * [~1*x + y]^2))))))")

lemma "w\<^sup>2 + x\<^sup>2 + y\<^sup>2 + z\<^sup>2 = 1 \ (w + x + y + z)\<^sup>2 \ (4::real)"
  by (sos "(((A<0 * R<1) + (([~4] * A=0) + (R<1 * ((R<3 * [~1/3*w + ~1/3*x + ~1/3*y + z]^2) + ((R<8/3 * [~1/2*w + ~1/2*x + y]^2) + (R<2 * [~1*w + x]^2)))))))")

lemma "(x::real) \ 1 \ y \ 1 \ x * y \ x + y - 1"
  by (sos "(((A<0 * R<1) + ((A<=0 * (A<=1 * R<1)) * (R<1 * [1]^2))))")

lemma "(x::real) > 1 \ y > 1 \ x * y > x + y - 1"
  by (sos "((((A<0 * A<1) * R<1) + ((A<=0 * R<1) * (R<1 * [1]^2))))")

lemma "\x\ \ 1 \ \64 * x^7 - 112 * x^5 + 56 * x^3 - 7 * x\ \ (1::real)"
  by (sos "((((A<0 * R<1) + ((A<=1 * R<1) * (R<1 * [~8*x^3 + ~4*x^2 + 4*x + 1]^2)))) & ((((A<0 * A<1) * R<1) + ((A<=1 * (A<0 * R<1)) * (R<1 * [8*x^3 + ~4*x^2 + ~4*x + 1]^2)))))")


text ‹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 "(((A<0 * R<1) + ((R<1 * ((R<5749028157/5000000000 * [~25000/222477*x + ~25000/222477*y + ~25000/222477*z + 1]^2) + ((R<864067/1779816 * [419113/864067*x + 419113/864067*y + z]^2) + ((R<320795/864067 * [419113/1283180*x + y]^2) + (R<1702293/5132720 * [x]^2))))) + (((A<=4 * (A<=5 * R<1)) * (R<3/2 * [1]^2)) + (((A<=3 * (A<=5 * R<1)) * (R<1/2 * [1]^2)) + (((A<=2 * (A<=4 * R<1)) * (R<1 * [1]^2)) + (((A<=2 * (A<=3 * R<1)) * (R<3/2 * [1]^2)) + (((A<=1 * (A<=5 * R<1)) * (R<1/2 * [1]^2)) + (((A<=1 * (A<=3 * R<1)) * (R<1/2 * [1]^2)) + (((A<=0 * (A<=4 * R<1)) * (R<1 * [1]^2)) + (((A<=0 * (A<=2 * R<1)) * (R<1 * [1]^2)) + ((A<=0 * (A<=1 * R<1)) * (R<3/2 * [1]^2)))))))))))))")


text ‹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 "((R<1 + ((R<1 * ((R<1 * [~1/6*x + ~1/6*y + ~1/6*z + 1]^2) + ((R<1/18 * [~1/2*x + ~1/2*y + z]^2) + (R<1/24 * [~1*x + y]^2)))) + (((A<0 * R<1) * (R<1/12 * [1]^2)) + (((A<=4 * (A<=5 * R<1)) * (R<1/6 * [1]^2)) + (((A<=2 * (A<=4 * R<1)) * (R<1/6 * [1]^2)) + (((A<=2 * (A<=3 * R<1)) * (R<1/6 * [1]^2)) + (((A<=0 * (A<=4 * R<1)) * (R<1/6 * [1]^2)) + (((A<=0 * (A<=2 * R<1)) * (R<1/6 * [1]^2)) + ((A<=0 * (A<=1 * R<1)) * (R<1/6 * [1]^2)))))))))))")


text ‹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 "(((A<0 * R<1) + (((A<=4 * R<1) * (R<2/3 * [1]^2)) + (((A<=4 * (A<=5 * R<1)) * (R<1 * [1]^2)) + (((A<=3 * (A<=4 * R<1)) * (R<1/3 * [1]^2)) + (((A<=2 * R<1) * (R<2/3 * [1]^2)) + (((A<=2 * (A<=5 * R<1)) * (R<1/3 * [1]^2)) + (((A<=2 * (A<=4 * R<1)) * (R<8/3 * [1]^2)) + (((A<=2 * (A<=3 * R<1)) * (R<1 * [1]^2)) + (((A<=1 * (A<=4 * R<1)) * (R<1/3 * [1]^2)) + (((A<=1 * (A<=2 * R<1)) * (R<1/3 * [1]^2)) + (((A<=0 * R<1) * (R<2/3 * [1]^2)) + (((A<=0 * (A<=5 * R<1)) * (R<1/3 * [1]^2)) + (((A<=0 * (A<=4 * R<1)) * (R<8/3 * [1]^2)) + (((A<=0 * (A<=3 * R<1)) * (R<1/3 * [1]^2)) + (((A<=0 * (A<=2 * R<1)) * (R<8/3 * [1]^2)) + ((A<=0 * (A<=1 * R<1)) * (R<1 * [1]^2))))))))))))))))))")


text ‹Inequality from sci.math (see "Leon-Sotelo, por favor").›

lemma "0 \ (x::real) \ 0 \ y \ x * y = 1 \ x + y \ x\<^sup>2 + y\<^sup>2"
  by (sos "(((A<0 * R<1) + (([1] * A=0) + (R<1 * ((R<1 * [~1/2*x + ~1/2*y + 1]^2) + (R<3/4 * [~1*x + y]^2))))))")

lemma "0 \ (x::real) \ 0 \ y \ x * y = 1 \ x * y * (x + y) \ x\<^sup>2 + y\<^sup>2"
  by (sos "(((A<0 * R<1) + (([~1*x + ~1*y + 1] * A=0) + (R<1 * ((R<1 * [~1/2*x + ~1/2*y + 1]^2) + (R<3/4 * [~1*x + y]^2))))))")

lemma "0 \ (x::real) \ 0 \ y \ x * y * (x + y)\<^sup>2 \ (x\<^sup>2 + y\<^sup>2)\<^sup>2"
  by (sos "(((A<0 * R<1) + (R<1 * ((R<1 * [~1/2*x^2 + y^2 + ~1/2*x*y]^2) + (R<3/4 * [~1*x^2 + x*y]^2)))))")

lemma "(0::real) \ a \ 0 \ b \ 0 \ c \ c * (2 * a + b)^3 / 27 \ x \ c * a\<^sup>2 * b \ x"
  by (sos "(((A<0 * R<1) + (((A<=3 * R<1) * (R<1 * [1]^2)) + (((A<=1 * (A<=2 * R<1)) * (R<1/27 * [~1*a + b]^2)) + ((A<=0 * (A<=2 * R<1)) * (R<8/27 * [~1*a + b]^2))))))")

lemma "(0::real) < x \ 0 < 1 + x + x\<^sup>2"
  by (sos "((R<1 + ((R<1 * (R<1 * [x]^2)) + (((A<0 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2))))))")

lemma "(0::real) \ x \ 0 < 1 + x + x\<^sup>2"
  by (sos "((R<1 + ((R<1 * (R<1 * [x]^2)) + (((A<=1 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2))))))")

lemma "(0::real) < 1 + x\<^sup>2"
  by (sos "((R<1 + ((R<1 * (R<1 * [x]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2)))))")

lemma "(0::real) \ 1 + 2 * x + x\<^sup>2"
  by (sos "(((A<0 * R<1) + (R<1 * (R<1 * [x + 1]^2))))")

lemma "(0::real) < 1 + \x\"
  by (sos "((R<1 + (((A<=1 * R<1) * (R<1/2 * [1]^2)) + ((A<=0 * R<1) * (R<1/2 * [1]^2)))))")

lemma "(0::real) < 1 + (1 + x)\<^sup>2 * \x\"
  by (sos "(((R<1 + (((A<=1 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [x + 1]^2))))) & ((R<1 + (((A<0 * R<1) * (R<1 * [x + 1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2))))))")


lemma "\(1::real) + x\<^sup>2\ = (1::real) + x\<^sup>2"
  by (sos "(() & (((R<1 + ((R<1 * (R<1 * [x]^2)) + ((A<1 * R<1) * (R<1/2 * [1]^2))))) & ((R<1 + ((R<1 * (R<1 * [x]^2)) + ((A<0 * R<1) * (R<1 * [1]^2)))))))")

lemma "(3::real) * x + 7 * a < 4 \ 3 < 2 * x \ a < 0"
  by (sos "((R<1 + (((A<1 * R<1) * (R<2 * [1]^2)) + (((A<0 * R<1) * (R<3 * [1]^2)) + ((A<=0 * R<1) * (R<14 * [1]^2))))))")

lemma "(0::real) < x \ 1 < y \ y * x \ z \ x < z"
  by (sos "((((A<0 * A<1) * R<1) + (((A<=1 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2)))))")

lemma "(1::real) < x \ x\<^sup>2 < y \ 1 < y"
  by (sos "((((A<0 * A<1) * R<1) + ((R<1 * ((R<1/10 * [~2*x + y + 1]^2) + (R<1/10 * [~1*x + y]^2))) + (((A<1 * R<1) * (R<1/2 * [1]^2)) + (((A<0 * R<1) * (R<1 * [x]^2)) + (((A<=0 * R<1) * ((R<1/10 * [x + 1]^2) + (R<1/10 * [x]^2))) + (((A<=0 * (A<1 * R<1)) * (R<1/5 * [1]^2)) + ((A<=0 * (A<0 * R<1)) * (R<1/5 * [1]^2)))))))))")

lemma "(b::real)\<^sup>2 < 4 * a * c \ a * x\<^sup>2 + b * x + c \ 0"
  by (sos "(((A<0 * R<1) + (R<1 * (R<1 * [2*a*x + b]^2))))")

lemma "(b::real)\<^sup>2 < 4 * a * c \ a * x^2 + b * x + c \ 0"
  by (sos "(((A<0 * R<1) + (R<1 * (R<1 * [2*a*x + b]^2))))")

lemma "(a::real) * x\<^sup>2 + b * x + c = 0 \ b\<^sup>2 \ 4 * a * c"
  by (sos "(((A<0 * R<1) + (R<1 * (R<1 * [2*a*x + b]^2))))")

lemma "(0::real) \ b \ 0 \ c \ 0 \ x \ 0 \ y \ x\<^sup>2 = c \ y\<^sup>2 = a\<^sup>2 * c + b \ a * c \ y * x"
  by (sos "(((A<0 * (A<0 * R<1)) + (((A<=2 * (A<=3 * (A<0 * R<1))) * (R<2 * [1]^2)) + ((A<=0 * (A<=1 * R<1)) * (R<1 * [1]^2)))))")

lemma "\x - z\ \ e \ \y - z\ \ e \ 0 \ u \ 0 \ v \ u + v = 1 \ \(u * x + v * y) - z\ \ (e::real)"
  by (sos "((((A<0 * R<1) + (((A<=3 * (A<=6 * R<1)) * (R<1 * [1]^2)) + ((A<=1 * (A<=5 * R<1)) * (R<1 * [1]^2))))) & ((((A<0 * A<1) * R<1) + (((A<=3 * (A<=5 * (A<0 * R<1))) * (R<1 * [1]^2)) + ((A<=1 * (A<=4 * (A<0 * R<1))) * (R<1 * [1]^2))))))")


lemma "(x::real) - y - 2 * x^4 = 0 \ 0 \ x \ x \ 2 \ 0 \ y \ y \ 3 \ y\<^sup>2 - 7 * y - 12 * x + 17 \ 0"
  oops (*Too hard?*)

lemma "(0::real) \ x \ (1 + x + x\<^sup>2) / (1 + x\<^sup>2) \ 1 + x"
  by (sos "(((((A<0 * A<1) * R<1) + ((A<=0 * (A<0 * R<1)) * (R<1 * [x]^2)))) & ((R<1 + ((R<1 * (R<1 * [x]^2)) + ((A<0 * R<1) * (R<1 * [1]^2))))))")

lemma "(0::real) \ x \ 1 - x \ 1 / (1 + x + x\<^sup>2)"
  by (sos "(((R<1 + (([~4/3] * A=0) + ((R<1 * ((R<1/3 * [3/2*x + 1]^2) + (R<7/12 * [x]^2))) + ((A<=0 * R<1) * (R<1/3 * [1]^2)))))) & (((((A<0 * A<1) * R<1) + ((A<=0 * (A<0 * R<1)) * (R<1 * [x]^2)))) & ((R<1 + ((R<1 * (R<1 * [x]^2)) + (((A<0 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2))))))))")

lemma "(x::real) \ 1 / 2 \ - x - 2 * x\<^sup>2 \ - x / (1 - x)"
  by (sos "((((A<0 * A<1) * R<1) + ((A<=0 * (A<0 * R<1)) * (R<1 * [x]^2))))")

lemma "4 * r\<^sup>2 = p\<^sup>2 - 4 * q \ r \ (0::real) \ x\<^sup>2 + p * x + q = 0 \ 2 * (x::real) = - p + 2 * r \ 2 * x = - p - 2 * r"
  by (sos "((((((A<0 * A<1) * R<1) + ([~4] * A=0))) & ((((A<0 * A<1) * R<1) + ([4] * A=0)))) & (((((A<0 * A<1) * R<1) + ([4] * A=0))) & ((((A<0 * A<1) * R<1) + ([~4] * A=0)))))")

end