Eisenstein no three trace zero extensions

import Mathlib.Algebra.QuadraticAlgebra.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Finset.Card
import Mathlib.Data.Int.Sqrt
import Mathlib.Tactic.Linarith
import Definitions.Def_D1Bound5_Coordinates_Definitions

section
namespace D1Bound5








theorem _root_.D1Bound5.eisenstein_no_three_trace_zero_extensions
    (x y z : QuadInt 1 1) (hx0 : x ≠ 0) (hy0 : y ≠ 0) (hz0 : z ≠ 0)
    (hx : 2 * x.re + x.im = 0) (hy : 2 * y.re + y.im = 0)
    (hz : 2 * z.re + z.im = 0)
    (hxy : ∃ r : QuadInt 1 1, r * r = x * y + 1)
    (hxz : ∃ r : QuadInt 1 1, r * r = x * z + 1)
    (hyz : ∃ r : QuadInt 1 1, r * r = y * z + 1) : False := by sorry



end D1Bound5
end