lean_workbook_plus_44018

⚠️ 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.

Put $\sqrt[3] {x}=a$ . The equation is equivalent to $a^2-2a-(a^3-4)\sqrt{a^3-7}-3a^3+28=0$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_44018 (x : ℝ) (hx : 0 < x) (a : ℝ) (ha : a = x^(1/3)) : a^2 - 2*a - (a^3 - 4)*Real.sqrt (a^3 - 7) - 3*a^3 + 28 = 0   :=  by sorry

Fractional-looking exponents elaborate as natural-number quotients, changing roots to powers0 (or another integer). The original equation in x was omitted, so a change of variables is asserted to satisfy an equation for every x.

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: Put $\sqrt[3] {x}=a$ . The equation is equivalent to $a^2-2a-(a^3-4)\sqrt{a^3-7}-3a^3+28=0$ Recover the original equation and prove an equivalence under a³=x and the source square-root domain; do not claim the transformed equation unconditionally.

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.