lean_workbook_plus_13140

⚠️ 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.

You get \n\n $\frac{a+b}{ab} \geq \frac{24}{a^{2}b^{2}+4ab+4}$ \n\nYou can get: \n\n $3a^{2}b^{2}+12ab+12 \geq 24ab$ \n\n $3a^{2}b^{2} + 12 \geq 12ab$ \n\n $4 \geq 4ab- a^{2}b^{2}$ \n\n $4 \geq ab(4-ab)$ \n\nThe maximum of the product of ab is $2.25$ . When you plug in $2.25$ , it comes out as: $3.9375$ \n\nSo the inequality is always true.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_13140 : ∀ a b : ℝ, (a + b) / (a * b) ≥ 24 / (a ^ 2 * b ^ 2 + 4 * a * b + 4)   :=  by sorry

The source derivation replaces a+b by 3, while the formal statement omits that normalization and domain.

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

Proposed corrected statement

Require a,b>0 and a+b=3 before asserting (a+b)/(ab)≥24/(ab+2)².

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.