lean_workbook_plus_76245

⚠️ 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.

Prove that $\frac{1}{2+x^2}+\frac{1}{2+y^2}+\frac{1}{2+z^2} \ge \frac{9}{10}$ given $x, y, z$ are nonnegative reals that satisfy $xy+yz+zx=3$.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_76245 x y z (hx: x >= 0 ∧ y >= 0 ∧ z >= 0) (hab : x*y + y*z + z*x = 3) : 1/(2 + x^2) + 1/(2 + y^2) + 1/(2 + z^2) >= 9/10   :=  by sorry

The untyped variables default to ℕ and all reciprocal terms and9/10 vanish.

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

Declare x,y,z:ℝ and perform all divisions over ℝ, preserving nonnegativity and xy+yz+zx=3. Source task: Prove that $\frac{1}{2+x^2}+\frac{1}{2+y^2}+\frac{1}{2+z^2} \ge \frac{9}{10}$ given $x, y, z$ are nonnegative reals that satisfy $xy+yz+zx=3$.

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.