Theory SOS

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

Examples for Sum_of_Squares.
*)

theory SOS
imports "HOL-Library.Sum_of_Squares"
begin

lemma "(3::real) * x + 7 * a < 4 ∧ 3 < 2 * x ⟹ a < 0"
  by sos

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

lemma "(3::real) * x + 7 * a < 4 ∧ 3 < 2 * x ⟶ a < 0"
  by sos

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

lemma "(0::real) ≤ x ∧ 0 ≤ y ∧ 0 ≤ z ∧ x + y + z ≤ 3 ⟶ x * y + x * z + y * z ≥ 3 * x * y * z"
  by sos

lemma "(x::real)⇧2 + y⇧2 + z⇧2 = 1 ⟶ (x + y + z)⇧2 ≤ 3"
  by sos

lemma "w⇧2 + x⇧2 + y⇧2 + z⇧2 = 1 ⟶ (w + x + y + z)⇧2 ≤ (4::real)"
  by sos

lemma "(x::real) ≥ 1 ∧ y ≥ 1 ⟶ x * y ≥ x + y - 1"
  by sos

lemma "(x::real) > 1 ∧ y > 1 ⟶ x * y > x + y - 1"
  by sos

lemma "¦x¦ ≤ 1 ⟶ ¦64 * x^7 - 112 * x^5 + 56 * x^3 - 7 * x¦ ≤ (1::real)"
  by sos


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


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


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


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

lemma "0 ≤ (x::real) ∧ 0 ≤ y ∧ x * y = 1 ⟶ x * y * (x + y) ≤ x⇧2 + y⇧2"
  by sos

lemma "0 ≤ (x::real) ∧ 0 ≤ y ⟶ x * y * (x + y)⇧2 ≤ (x⇧2 + y⇧2)⇧2"
  by sos

lemma "(0::real) ≤ a ∧ 0 ≤ b ∧ 0 ≤ c ∧ c * (2 * a + b)^3 / 27 ≤ x ⟶ c * a⇧2 * b ≤ x"
  by sos

lemma "(0::real) < x ⟶ 0 < 1 + x + x⇧2"
  by sos

lemma "(0::real) ≤ x ⟶ 0 < 1 + x + x⇧2"
  by sos

lemma "(0::real) < 1 + x⇧2"
  by sos

lemma "(0::real) ≤ 1 + 2 * x + x⇧2"
  by sos

lemma "(0::real) < 1 + ¦x¦"
  by sos

lemma "(0::real) < 1 + (1 + x)⇧2 * ¦x¦"
  by sos


lemma "¦(1::real) + x⇧2¦ = (1::real) + x⇧2"
  by sos

lemma "(3::real) * x + 7 * a < 4 ∧ 3 < 2 * x ⟶ a < 0"
  by sos

lemma "(0::real) < x ⟶ 1 < y ⟶ y * x ≤ z ⟶ x < z"
  by sos

lemma "(1::real) < x ⟶ x⇧2 < y ⟶ 1 < y"
  by sos

lemma "(b::real)⇧2 < 4 * a * c ⟶ a * x⇧2 + b * x + c ≠ 0"
  by sos

lemma "(b::real)⇧2 < 4 * a * c ⟶ a * x⇧2 + b * x + c ≠ 0"
  by sos

lemma "(a::real) * x⇧2 + b * x + c = 0 ⟶ b⇧2 ≥ 4 * a * c"
  by sos

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

lemma "¦x - z¦ ≤ e ∧ ¦y - z¦ ≤ e ∧ 0 ≤ u ∧ 0 ≤ v ∧ u + v = 1 ⟶ ¦(u * x + v * y) - z¦ ≤ (e::real)"
  by sos

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; left it running for 80 minutes -- LCP*)

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

lemma "(0::real) ≤ x ⟶ 1 - x ≤ 1 / (1 + x + x⇧2)"
  by sos

lemma "(x::real) ≤ 1 / 2 ⟶ - x - 2 * x⇧2 ≤ - x / (1 - x)"
  by sos

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

end