lean_workbook_plus_71766

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Let $x,y,z$ be non-negative real number,prove that:\n$2\sum{\sqrt{(x+y+4z)(y+z+4x)}}\ge{9\sum{\frac{(x+y)(x+z)+(x+\frac{y+z}{2})^2}{2x+y+z}}-6(x+y+z)}$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_71766 : ∀ x y z : ℝ, 2 * (Real.sqrt ((x + y + 4 * z) * (y + z + 4 * x)) + Real.sqrt ((x + y + 4 * z) * (x + z + 4 * y)) + Real.sqrt ((x + z + 4 * y) * (y + z + 4 * x))) ≥ 9 * ((x + y) * (x + z) + (x + (y + z) / 2) ^ 2 / (2 * x + y + z) + (y + z) * (y + x) + (y + (x + z) / 2) ^ 2 / (2 * y + x + z) + (z + x) * (z + y) + (z + (x + y) / 2) ^ 2 / (2 * z + x + y)) / 2 - 6 * (x + y + z)   :=  by sorry

Numerators were only partially divided, a global factor1/2 was inserted, and source nonnegativity was omitted.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

For x,y,z≥0 restore each cyclic fraction ((x+y)(x+z)+(x+(y+z)/2)²)/(2x+y+z), sum them, multiply by9 and subtract6(x+y+z). Treat the all-zero case separately or require a positive sum.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.