lean_workbook_plus_82423

⚠️ 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.

Prove that for all reals $a$, $b$, and $c$, if $k \geq \sqrt[3]{\frac{27}{16}}$, then $\sqrt[3]{ka^3-abc}+\sqrt[3]{kb^3-abc}+\sqrt[3]{kc^3-abc}\geq 0$.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_82423 (a b c k : ℝ) (h₁ : k ≥ (27 / 16)^(1 / 3)) : (k * a ^ 3 - a * b * c) ^ (1 / 3) + (k * b ^ 3 - a * b * c) ^ (1 / 3) + (k * c ^ 3 - a * b * c) ^ (1 / 3) ≥ 0   :=  by sorry

The natural exponents erase every cube root. The all-real source is additionally false: a=b=c=−1 and k=2 satisfy k≥∛(27/16), but the signed-root sum is−3.

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 exponents carry no type ascription, so they elaborate at type ℕ where division truncates (^ (1 / 3) → ^0; ^(1 / 3) → ^0).

Proposed corrected statement

Use genuine signed cube roots and determine exactly which domain/normalization on a,b,c was intended. The quoted all-real universal claim must be replaced by a counterexample problem (a=b=c=−1,k=2), or supplied with the actual omitted source restrictions; do not assume a root-typing repair alone proves it.

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.