lean_workbook_plus_9441

⚠️ 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.

prove that: \n $\frac{1}{x+x^2}+\frac{1}{y+y^2}+\frac{1}{z+z^2} \geq \frac{6}{-5+3(x+y+z)}$ \n best is: \n $x,y,z\geq 0,x^2+y^2+z^2=3$ ,

Why this node was retired

The posted statement is

theorem lean_workbook_plus_9441 : ∀ x y z : ℝ, (x^2 + y^2 + z^2 = 3 ∧ x >= 0 ∧ y >= 0 ∧ z >= 0) ∨ (x^2 + y^2 + z^2 < 3) → 1/(x + x^2) + 1/(y + y^2) + 1/(z + z^2) >= 6/(-5 + 3 * (x + y + z))   :=  by sorry

An extra disjunct x²+y²+z²<3 admits points outside the source constraint; the source also needs nonzero reciprocal coordinates.

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

Proposed corrected statement

Remove the disjunct, require positive x,y,z with x²+y²+z²=3, and verify the proposed reciprocal inequality and denominator sign.

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.