lean_workbook_plus_4908

⚠️ 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.

Determine the smallest and the greatest possible values of the expression \n\n

provided $a,b$ and $c$ are non-negative real numbers satisfying $ab+bc+ca=1$ .

Why this node was retired

The posted statement is

theorem lean_workbook_plus_4908 (a b c : ℝ) (hab : a * b + b * c + c * a = 1) : 2 ≤ ((1 / (a ^ 2 + 1) + 1 / (b ^ 2 + 1) + 1 / (c ^ 2 + 1)) * (a ^ 2 / (a ^ 2 + 1) + b ^ 2 / (b ^ 2 + 1) + c ^ 2 / (c ^ 2 + 1))) ∧ ((1 / (a ^ 2 + 1) + 1 / (b ^ 2 + 1) + 1 / (c ^ 2 + 1)) * (a ^ 2 / (a ^ 2 + 1) + b ^ 2 / (b ^ 2 + 1) + c ^ 2 / (c ^ 2 + 1))) ≤ 27 / 16   :=  by sorry

The target’s lower bound2 exceeds its upper bound27/16.

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

Proposed corrected statement

Restore a,b,c≥0 and reverse the invented endpoints:27/16≤expression≤2. For extrema, include attainability at the equal triple and at permutations of(1,1,0).

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.