lean_workbook_plus_31463

⚠️ 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.

For all $a\ge13$ , prove the inequality; $\sqrt[3]{1-\frac{1}{a}}+\sqrt[3]{1+\frac{1}{a}}<2$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_31463 : ∀ a ≥ 13, (1 - 1 / a)^(1 / 3) + (1 + 1 / a)^(1 / 3) < 2   :=  by sorry

Fractional-looking exponents elaborate as natural-number quotients, changing roots to powers0 (or another integer).

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 exponent carries no type ascription, so it elaborates at type ℕ where division truncates (^(1 / 3) → ^0).

Proposed corrected statement

Use genuine real roots and explicitly real fractional exponents. For odd roots of negative values use the signed real root. Retain the source domain and verify the repaired claim rather than infer its truth from this counterexample. Original task: For all $a\ge13$ , prove the inequality; $\sqrt[3]{1-\frac{1}{a}}+\sqrt[3]{1+\frac{1}{a}}<2$ Type a asℝ so the fractions1/a are genuine.

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.