lean_workbook_plus_36839

⚠️ 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 $a,b,c$ be non-negative numbers, such that $a + b + c + abc = 4$ Prove that: $3abc \le ab + bc + ca \le abc + 4$ The second part can be strengthened. Actually, the stronger inequality holds: If $a,$ $b,$ $c$ are nonnegative real numbers such that $a + b + c + abc = 4,$ then $ab+bc+ca \le a+b+c.$ This inequality is equivalent to Vietnamese TST 1996 problem.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_36839 : ∀ a b c : ℝ, a + b + c + a * b * c = 4 → 3 * a * b * c ≤ a * b + b * c + c * a ∧ a * b + b * c + c * a ≤ a * b * c + 4   :=  by sorry

Source nonnegativity was omitted.

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 alongside a+b+c+abc=4.

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.