lean_workbook_plus_64760

⚠️ 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.

We have $\begin{array}{l}\frac{{ab}}{{ab + c}} + \frac{{bc}}{{bc + a}} + \frac{{ca}}{{ca + b}} \le \frac{3}{2}\\\Leftrightarrow \frac{c}{{ab + c}} + \frac{a}{{bc + a}} + \frac{b}{{ca + b}} \ge \frac{3}{2}\\\Leftrightarrow \sum {{a^2}{b^2}} + 3abc \ge abc\left( {a + b + c} \right) + 3{a^2}{b^2}{c^2}\end{array}$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_64760 : ∀ a b c : ℝ, (a * b / (a * b + c) + b * c / (b * c + a) + c * a / (c * a + b) ≤ 3 / 2 ↔ c / (a * b + c) + a / (b * c + a) + b / (c * a + b) ≥ 3 / 2)   :=  by sorry

The complementary fractions require nonzero common denominators; all-zero input destroys the equivalence.

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

Proposed corrected statement

Assume ab+c,bc+a,ca+b are all nonzero. Then each complementary pair sums to1, proving the displayed equivalence.

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.