lean_workbook_plus_8223

⚠️ 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.

$\frac{x^2}{(x^2y)^{\frac{1}{3}}}+\frac{y^2}{(y^2z)^{\frac{1}{3}}}+\frac{z^2}{(z^2x)^{\frac{1}{3}}}\ge\frac{x^2+y^2+z^2}{({\frac{x^2y*x^2+y^2z*y^2+z^2x*z^2}{x^2+y^2+z^2}})^{\frac{1}{3}}}$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_8223 (x y z : ℝ) :
  x ^ 2 / (x ^ 2 * y) ^ (1 / 3) + y ^ 2 / (y ^ 2 * z) ^ (1 / 3) + z ^ 2 / (z ^ 2 * x) ^ (1 / 3) ≥
    (x ^ 2 + y ^ 2 + z ^ 2) / (x ^ 2 * y + y ^ 2 * z + z ^ 2 * x) ^ (1 / 3)   :=  by sorry

All cube-root denominators become 1; the right radicand also loses the source’s weighted average and its normalization.

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

Restore the denominator cbrt((x⁴y+y⁴z+z⁴x)/(x²+y²+z²)) on the right and cbrt(x²y),cbrt(y²z),cbrt(z²x) on the left. Recover x,y,z>0 for the reciprocal-convexity argument.

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.