lean_workbook_plus_15900

⚠️ 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 $a = \frac {1}{x}, b = \frac {1}{y}, c = \frac {1}{z}$ , we have $a + b + c = 1$ and \n\n $(1 - \frac {1}{1 + x^2})(1 - \frac {1}{1 + y^2})(1 - \frac {1}{1 + z^2}) > \frac {1}{2}$ \n\n $\Leftrightarrow 2 > (a^2 + 1)(b^2 + 1)(c^2 + 1)$ .

Why this node was retired

The posted statement is

theorem lean_workbook_plus_15900 :
  ∀ x y z : ℝ, 1 / x + 1 / y + 1 / z = 1 → (1 - 1 / (1 + x^2)) * (1 - 1 / (1 + y^2)) * (1 - 1 / (1 + z^2)) > 1 / 2   :=  by sorry

The excerpt is a reformulation step, and the target admits undefined reciprocal boundary values.

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

Proposed corrected statement

Require x,y,z>0 for the claimed strict inequality, or state only the source’s equivalence after the reciprocal substitution with nonzero inputs.

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.