lean_workbook_plus_39699

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Proved status carries no information about that problem. Do not import this node or use it as a dependency.

Let $a=3+x,b=3+y,c=3+z,x,y,z\ge 0$ .Then the given inequality is equivalent to $3xyz+7xy+5xz+3yz+9x+6y+3z\ge 0$ ,which is obviously true.We have equality if and only if $x=y=z=0$ i.e. $a=b=c=3$ .

Why this node was retired

The posted statement is

theorem lean_workbook_plus_39699 (a b c x y z : ℝ) (hx : x ≥ 0) (hy : y ≥ 0) (hz : z ≥ 0) (hab : a = 3 + x) (hbc : b = 3 + y) (hca : c = 3 + z) : a * b * c ≥ 3 * a + 3 * b + 3 * c - 9 ↔ 3 * x * y * z + 7 * x * y + 5 * x * z + 3 * y * z + 9 * x + 6 * y + 3 * z ≥ 0   :=  by sorry

The target substitutes a different left inequality whose gap at x=y=z=0 is 9, so its equality cases cannot match the source. The accepted biconditional only proves two unrelated positive statements.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

Recover the original inequality and prove its actual cleared difference equals the displayed nonnegative polynomial; state equality iff x=y=z=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.