lean_workbook_plus_25134

⚠️ 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.

Lemma: $\frac{x}{1-x} \geq \frac{9+6\sqrt{3}}{4} x^2 - \frac{1}{4} \Leftrightarrow (x-\frac{1}{\sqrt{3}})^2\cdot ((2\sqrt{3}+3)x + 1) \geq 0$ So: $\frac{a}{1-a} + \frac{b}{1-b} + \frac{c}{1-c} \geq \frac{9+6\sqrt{3}}{4}(a^2+b^2+c^2) - \frac{3}{4} = \frac{9+6\sqrt{3}}{4}\cdot 1 - \frac{3}{4} = \frac{6+6\sqrt{3}}{4} = \frac{3+3\sqrt{3}}{2}$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_25134 : ∀ x : ℝ, (x / (1 - x) ≥ (9 + 6 * Real.sqrt 3) / 4 * x ^ 2 - 1 / 4 ↔ (x - 1 / Real.sqrt 3) ^ 2 * ((2 * Real.sqrt 3 + 3) * x + 1) ≥ 0)   :=  by sorry

Require x<1 so multiplication by1−x is positive; within that domain the displayed polynomial sign equivalence can be justified.

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 x<1 so multiplication by1−x is positive; within that domain the displayed polynomial sign equivalence can be justified.

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.