lean_workbook_plus_16231

⚠️ 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.

Try the following estimation \n $\sum _{cyc}\left(\frac {a}{a + b}\right)^2 = \sum _{cyc}\frac {a^2(a + c)^2}{(a + c)^2(a + b)^2}\geq\frac {\left(\sum(a^2 + ab)\right)^2}{\sum(a + b)^2(a + c)^2}.$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_16231 :
  ∀ a b c : ℝ,
    (a / (a + b))^2 + (b / (b + c))^2 + (c / (c + a))^2 ≥
    (a^2 + a * b + b^2 + b * c + c^2 + c * a)^2 / ((a + b)^2 * (b + c)^2 + (b + c)^2 * (c + a)^2 + (c + a)^2 * (a + b)^2)   :=  by sorry

The rewritten fractions cancel pair sums without excluding their zeros; the counterexample has c+a=0.

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,b+c,c+a all nonzero, then verify the stated Cauchy–Schwarz inequality after matching cyclic factors; positive a,b,c is a sufficient candidate domain.

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.