lean_workbook_plus_46741

⚠️ 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.

Given $a, b, c > 0$, prove that\n$\sum\limits_{cyc} \left(\frac{a}{a + b + c} - \frac{1}{3}\right)\left(\frac{a}{a + b} - \frac{1}{2}\right) \ge \frac{2}{3} \left(\sum\limits_{cyc} \frac{b + c}{2a + b + c} - \frac{3}{2}\right)$\n\nis equivalent to\n\n$\sum\limits_{cyc} a(2a^2b^2 + 5a^2bc + 3a^2c^2 + 10ab^3 + 10acb^2 + 5abc^2 + 27ac^3 + 2b^4 + 32c^3b)(a - b)^2 \ge 0$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_46741 :
  ∀ a b c : ℝ, a > 0 ∧ b > 0 ∧ c > 0 →
    (a / (a + b + c) - 1 / 3) * (b / (b + c) - 1 / 2) +
    (b / (b + c + a) - 1 / 3) * (c / (c + a) - 1 / 2) +
    (c / (c + a + b) - 1 / 3) * (a / (a + b) - 1 / 2) ≥
    2 / 3 * ((b + c) / (2 * a + b + c) + (c + a) / (2 * b + c + a) + (a + b) / (2 * c + a + b) - 3 / 2)   :=  by sorry

The second factor of the first source summand is a/(a+b)−1/2, but the target uses b/(b+c)−1/2 and shifts each cyclic term.

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

Proposed corrected statement

Restore the source indexing in each product, and separately formalize its claimed polynomial equivalence if desired.

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.