lean_workbook_plus_72639

⚠️ 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.

Let $a,b,c,d$ are non-negative real numbers such that $a^2 -ac + c^2 = b^2 - bd + d^2$ . $(a) \frac{1}{a^2 + c^2} + \frac{1}{b^2 + d^2} \le \frac{4}{(a+c)(b+d)}$ $(b) \frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{c+d} + \frac{1}{d+a} \le \frac{a+b+c+d}{ac+bd}$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_72639 : ∀ a b c d : ℝ, a ≥ 0 ∧ b ≥ 0 ∧ c ≥ 0 ∧ d ≥ 0 ∧ a^2 - a * c + c^2 = b^2 - b * d + d^2 → 1 / (a^2 + c^2) + 1 / (b^2 + d^2) ≤ 4 / (a + c) / (b + d) ∧ 1 / (a + b) + 1 / (b + c) + 1 / (c + d) + 1 / (d + a) ≤ (a + b + c + d) / (a * c + b * d)   :=  by sorry

The source nonnegative domain includes undefined divisions at boundary points.

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 the denominators in both inequalities to be nonzero, or restrict all four variables to be strictly positive, before using ordinary reciprocal arithmetic.

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.