# Theory SOS_Cert

theory SOS_Cert
imports Sum_of_Squares
```(*  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)⇧2 + y⇧2 + z⇧2 = 1 ⟶ (x + y + z)⇧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⇧2 + x⇧2 + y⇧2 + z⇧2 = 1 ⟶ (w + x + y + z)⇧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⇧2 + y⇧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⇧2 + y⇧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)⇧2 ≤ (x⇧2 + y⇧2)⇧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⇧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⇧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⇧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⇧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⇧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)⇧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⇧2¦ = (1::real) + x⇧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⇧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)⇧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 "(b::real)⇧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⇧2 + b * x + c = 0 ⟶ b⇧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⇧2 = c ∧ y⇧2 = a⇧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⇧2 - 7 * y - 12 * x + 17 ≥ 0"
oops (*Too hard?*)

lemma "(0::real) ≤ x ⟶ (1 + x + x⇧2) / (1 + x⇧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⇧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⇧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⇧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 "((((((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
```