lean_workbook_plus_18802

⚠️ 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.

Prove that $\forall x,y,z \in (0, \infty)$ we have $\sqrt[2]{\sqrt[3]{x} + \sqrt[3]{y} + \sqrt[3]{z}} > \sqrt[3]{\sqrt[2]{x} + \sqrt[2]{y} + \sqrt[2]{z}}$ .

Why this node was retired

The posted statement is

theorem lean_workbook_plus_18802 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : ((x + y + z) ^ (1 / 2) : ℝ) > (x ^ (1 / 3) + y ^ (1 / 3) + z ^ (1 / 3)) ^ (1 / 3)   :=  by sorry

Fractional-looking exponents elaborate as natural-number quotients, changing roots to powers0 (or another integer). Both nested radical expressions were replaced by different sums.

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

Confirmed directly from the Lean text: the numeric-literal exponents carry no type ascription, so they elaborate at type ℕ where division truncates (^ (1 / 2) → ^0; ^ (1 / 3) → ^0).

Proposed corrected statement

For positive x,y,z compare √(∛x+∛y+∛z) with∛(√x+√y+√z), using genuine nested roots as in the source.

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.