lean_workbook_plus_29555

⚠️ Retired — specification defect

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

Let $a$ , $b$ , $c$ , $k$ be positive reals such that $ab+bc+ca \leq 1$ and $0 < k \leq \frac{9}{2}$ . Prove that:\n\n$\sqrt[3]{ \frac{k}{a} + (9-3k)b} + \sqrt[3]{\frac{k}{b} + (9-3k)c} + \sqrt[3]{\frac{k}{c} + (9-3k)a } \leq \frac{1}{abc}.$\nProposed by Zhang Yanzong and Song Qing

Why this node was retired

The posted statement is

theorem lean_workbook_plus_29555 (a b c k : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a * b * c = 1) (h : a * b + b * c + c * a ≤ 1) (hk : 0 < k ∧ k ≤ 9 / 2) :
  ( k / a + (9 - 3 * k) * b)^(1 / 3) + ( k / b + (9 - 3 * k) * c)^(1 / 3) + ( k / c + (9 - 3 * k) * a)^(1 / 3) ≤ 1 / (a * b * c)   :=  by sorry

The added abc=1 contradicts ab+bc+ca≤1 for positive variables, and all cube-root exponents become0.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Confirmed directly from the Lean text: the numeric-literal exponent carries no type ascription, so it elaborates at type ℕ where division truncates (^(1 / 3) → ^0).

Proposed corrected statement

Remove the extra normalization; retain exactly the positivity/domain restrictions and the displayed source constraint, then prove the displayed source conclusion. Source task: Let $a$ , $b$ , $c$ , $k$ be positive reals such that $ab+bc+ca \leq 1$ and $0 < k \leq \frac{9}{2}$ . Prove that:\n\n$\sqrt[3]{ \frac{k}{a} + (9-3k)b} + \sqrt[3]{\frac{k}{b} + (9-3k)c} + \sqrt[3]{\frac{k}{c} + (9-3k)a } \leq \frac{1}{abc}.$\nProposed by Zhang Yanzong and Song Qing Use signed real cube roots if a radicand can be negative; verify radicand signs from the exact source constraints instead of assuming them.

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.